on the crosscorrelation of polyphase power residue...

On the Crosscorrelation of Polyphase

Power Residue Sequences

Young-Joon Kim

The Graduate School

Yonsei University

Department of Electrical and Electronic

Engineering

On the Crosscorrelation of PolyphasePower Residue Sequences

A Dissertation

Submitted to the

Department of Electrical and Eletronic Engineering

and the Graduate School of Yonsei University

in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE

Young-Joon Kim

December 2003

This certifies that the dissertationof Young-Joon Kim is approved.

Thesis Supervisor: Hong-Yeop Song

DaeSik Hong

ChungYong Lee

The Graduate SchoolYonsei UniversityDecember 2003

Contents

List of Tables iii

Abstract iv

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory of Polyphase Power Residue Sequences 3

2.1 Legendre Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Polyphase Power Residue Sequences . . . . . . . . . . . . . . . . . . . 5

2.2.1 Generation of Polyphase Power Residue Sequences . . . . . . . 5

2.2.2 Autocorrelation of Polyphase Power Residue Sequences . . . . 7

3 Crosscorrelation of Polyphase Power Residue Sequences 11

3.1 Set of distinct q-phase Power Residue Sequence . . . . . . . . . . . . . 11

3.2 Crosscorrelation of two q-phase PRS of length p . . . . . . . . . . . . . 21

4 Transformation of q-ary PRS and their crosscorrelations 27

i

5 Concluding Remarks 35

Appendix 37

Bibliography 64

ii

List of Tables

3.1 All the q-ary PRS of length p up to 23 . . . . . . . . . . . . . . . . . . 18

3.2 Sets of exponents of µ that gives the same q-ary PRS of length p ≤ 73. . 19

3.3 Sets of exponents of µ that gives the same q-ary PRS of length p ≤ 73

(continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Maximum Crosscorrelation of PRS’s of length p ≤ 19. . . . . . . . . . 23

3.5 Maximum and the number of levels of Crosscorrelations of PPRS. . . . 24



4.1 Crosscorrelation of two distict q-ary sequence in A(M(s(n))) . . . . . 31

4.2 Maximum Crosscorrelation of decimations of 5-ary PRS of length 11 . 34

5.1 Maximum Crosscorrelation of PRS’s of length 23 ≤ p ≤ 53. . . . . . . 37

5.2 Two-level Crosscorrelation pairs in A(M(s(n))) p ≤ 97. . . . . . . . . 45

5.3 Crosscorrelations of D(M(s(n))) for p ≤ 97. . . . . . . . . . . . . . . 57

iii

ABSTRACT

On the Crosscorrelation of Polyphase Power ResidueSequences

Young-Joon KimDepartment of Electricaland Electronic Eng.The Graduate SchoolYonsei University

In the systems such as ranging, radar, and spread-spectrum communication systems, it

needs to find the sequences with a good correlation property in order to improve the per-

formance. The correlation property may be auto-correlation or crosscorrelation accord-

ing to the application. Like binary Legendre Sequences or quadratic residue sequences

which have ideal autocorrealtion, Polyphase Power Residue Sequence(shortly PPRS)

also have good autocorrelation. In this thesis, we make the set of PPRS by changing a

primitive root and the set of q-ary sequences through various transformations for a q-ary

PRS, and we compute the crosscorrelations of these sequences. The constant multiple,

the affine shift, the cyclic shift and the decimation are considered as these transforma-

tions. We show that the set of all the sequences through the constant multiple is the same

iv

with the set of all the sequences through changing a primitive root. We compute the

crosscorrelation Ca,b of two distinct q-phase PRS a(n) and b(n) in these set. From the

computing, we suggest that |Ca,b| is upper bounded to√

p + 2. Next we make the set

of sequences by considering the affine shift and the constant multiple simultaneously.

We collect the pairs giving two level crosscorrelation. We observe that the maximum

crosscorrelation of these pairs is exactly√

p. Finally, we consider the decimation and

the constant multiple as the transform. When we apply only the decimation to a q-ary

PRS as the transform, the crosscorrelation is much larger than√

p and what is worse, the

crosscorrelation is equal to or larger than p − 2. On the other hand, when we consider

both the constant multiple and the decimation, an interesting point is observed. If two

sequences are from the distinct q-ary PRS, then the crosscorrelation is upper bounded to

√p + 2 regardless of the decimation.

Key words : Legendre Sequences, Polyphase Power Residue Sequences, autocorre-lation, crosscorrelation

v

Chapter 1

Introduction

1.1 Motivation

It is well known that binary Legendre Sequences or quadratic residue sequences have

ideal autocorrelation functions when the length p ≡ 3 (mod 4) [1]. This type of se-

quences can be generalized to the muliple valued cases [2] [3]. Legendre Sequences is

a special case of q-phase power residue sequences with q=2. In general, the sequences

used in the spread spectrum communication need a good crosscorrelation function as

well as a good autocorrelation function for the discrimination of users [4] [5]. Kasami

and Gold generated the binary sequence families having the excellent crosscorrelation

properties by using m-sequences [6] [7]. Legendre sequences have been studied in terms

of the generation [8] [1], existence conditions [9] [10], the linear complexity [11] and

the trace representation [12] etc. On the other hand, only the autocorrelation and the

linear complexity of polyphase power residue sequences have been determined [2]. In

this thesis we make q-phase PRS set and investigate the crosscorrelations between two

distinct PPRS in the set.

1

1.2 An Overview

In Chapter 2, we review major properties of Legendre sequences and polyphase power

residue sequences. We also review a well-known construction of polyphase power

residue sequences. Chapter 3 discusses how to construct sets of PPRS of the same length.

Some important equivalence relations are described. Through the computing the cross-

correlaton of the above PPRS set, we suggest an interesting conjecture. In Chapter 4,

various transformations are proposed to construct larger sets, and we generate the q-ary

sequences set from these transformations. Crosscorrelation properties of the above sets

of PPRS are calculated with computers, and some interesting conjectures are obtained.

Finally in Chapter 5, all those results of this thesis are summarized and some discussions

follow.

2

Chapter 2

Theory of Polyphase Power ResidueSequences

Let a(n) be a binary {0, 1}- sequence of period p. Then the periodic autocorrelation of

a(n) is defined by

C(τ) =p−1∑x=0

(−1)a(x)−a(x+τ) (2.1)

2.1 Legendre Sequences

Legendre sequence {a(n)} of period p where p is an odd prime is defined as

a(n) =

1, if n ≡ 0 (mod p)

0, if n is a quadratic residue (mod p)

1, if n is a quadratic non-residue (mod p)

If p ≡ 3 (mod 4), the corresponding Legendre sequence not only is balanced but

also has the optimal autocorrelation property. Legendre sequences of period p ≡ 1 (mod

4) do not have the ideal autocorrelation property. But their autocorrelation property is

regarded to be good since the maximum amplitude of unnormalized out-of-phase auto-

correlation value is just 3. Here, we give an example which depicts the construction and

3

the autocorrelation property of Legendre sequences.

Example 2.1 Legendre sequence of period 11 ≡ 3 (mod 4) : The integer 2 is a primi-

tive root (mod 11) and the successive powers of 2 (mod 11) are given by :

i 0 1 2 3 4 5 6 7 8 9

2i (mod 11) 1 2 4 8 5 10 9 7 3 6

From the definition of Legendre sequences,

i 0 1 2 3 4 5 6 7 8 9 10

s(i) 1 0 1 0 0 0 1 1 1 0 1

And the autocorrelation of Legendre sequence(length=11)

τ 0 1 2 3 4 5 6 7 8 9 10

C(τ) 11 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Example 2.2 Legendre sequence of period 13 ≡ 1 (mod 4): The integer 2 is a primi-

tive root (mod 13) and the successive powers of 2 (mod 13) are given by :

i 0 1 2 3 4 5 6 7 8 9 10 11

2i(mod 13) 1 2 4 8 3 6 12 11 9 5 10 7

From the definition of Legendre sequences,

i 0 1 2 3 4 5 6 7 8 9 10 11 12

s(i) 1 0 1 0 0 1 1 1 1 0 0 1 0

And the autocorrelation of Legendre sequence(length=13)

τ 0 1 2 3 4 5 6 7 8 9 10 11 12

C(τ) 13 1 -3 1 1 -3 -3 -3 -3 1 1 -3 1

4

2.2 Polyphase Power Residue Sequences

2.2.1 Generation of Polyphase Power Residue Sequences

Let p be an odd prime and q be a divisor of p − 1. Let T = (p − 1)/q and µ be a

primitive root mod p. The nonzero integers mod p can be partitioned into q cosets Ci,

0 ≤ i ≤ q − 1, where C0 is the set of the q-th power residues mod p, and the remaining

Ci are formed as µi · C0.

Each Ci has exactly T elements. It is because that C0 has T elements and the size

of Ci is the same with that of C0 for 1 ≤ i ≤ q − 1.

Then q-ary power residue sequences {s(n)} taking values on Zq is the sequences defined

by the following rule. The values of {s(n)} are determined by which coset Ck contains

the sequence index n. In other words,

s(n) =

s(0), if n ≡ 0 mod p

0, if n ∈ C0

1, if n ∈ C1

...

q − 1, if n ∈ Cq−1

where n = 0, 1, ..., p − 1 and s(0) takes any q available values of Zq. Unless stated

otherwise, assume s(0) = 0.

Now consider the complex sequences {a(n)} which is the complex version of q-ary se-

quences {s(n)}. Let w be a complex primtive q-th root of unity or w = exp(j 2πq ).

Then q-ary sequences {s(n)} can be converted to complex sequences {a(n)} by a(n) =

ws(n) [2] [3]. In other words,

5

a(n) =

1, if n ≡ 0 mod p

w0 = 1, if n ∈ C0

w1, if n ∈ C1

...

wq−1, if n ∈ Cq−1

where n = 0, 1, ..., p − 1 and w = exp(j 2πq ).

For the convenience, we will call polypase power residue sequence with PPRS and

q-phase power residue sequence with q-phase PRS. We will denote {s(n)} the q-ary

PRS and {a(n)} the q-phase PRS or the complex version of {s(n)}.

Example 2.3 if p=13, q=3, µ=2,

i 0 1 2 3 4 5 6 7 8 9 10 11

2i(mod 13) 1 2 4 8 3 6 12 11 9 5 10 7

i 1 2 3 4 5 6 7 8 9 10 11 12

i3(mod 13) 1 8 1 12 8 8 5 5 1 12 5 12

Hence

C0 = {1, 5, 8, 12}

C1 = µ1 · C0 = 2 · C0 = 2 · {1, 5, 8, 12} = {2, 10, 3, 11}

C2 = µ2 · C0 = 22 · C0 = 4 · {1, 5, 8, 12} = {4, 7, 6, 9}

So 3-ary PRS {s(n)} and 3-phase PRS {a(n)} of length 13 is given as follows.

n 0 1 2 3 4 5 6 7 8 9 10 11 12

s(n) 0 0 1 1 2 0 2 2 0 2 1 1 0

a(n) 1 1 w1 w1 w2 1 w2 w2 1 w2 w1 w1 1

where w = exp(j 2π3 ).

6

Example 2.4 if p=13, q=4, µ=2,

i 1 2 3 4 5 6 7 8 9 10 11 12

i4(mod 13) 1 3 3 9 1 9 9 1 9 3 3 1

Hence,

C0 = {1, 3, 9}

C1 = µ1 · C0 = 2 · C0 = 2 · {1, 3, 9} = {2, 6, 5}

C2 = µ2 · C0 = 22 · C0 = 4 · {1, 3, 9} = {4, 12, 10}

C3 = µ3 · C0 = 23 · C0 = 8 · {1, 3, 9} = {8, 11, 7}

So 4-ary PRS {s(n)} and 4-phase PRS {a(n)} of length 13 is given as follows.

n 0 1 2 3 4 5 6 7 8 9 10 11 12

s(n) 0 0 1 0 2 1 1 3 3 0 2 3 2

a(n) 1 1 w1 1 w2 w1 w1 w3 w3 1 w2 w3 w2

where w = exp(j 2π4 ).

2.2.2 Autocorrelation of Polyphase Power Residue Sequences

The following properties of q-phase PRS {a(n)} with length p were derived previ-

ously [3]. Note that {a(n)} is the complex version of q-ary PRS {s(n)} .i.e. a(n) =

ws(n).

• a(u) · a(v) = a(u · v) (u �= 0, v �= 0)

• a(u) · a(v)∗ = a(u/v) (u �= 0, v �= 0)

7

• a(1) = 1 and

a(−1) =

{−1, if p ≡ q + 1 (mod 2q)

1, if p ≡ 1 (mod 2q)

a(−u) =

{−a(u), if p ≡ q + 1 (mod 2q)

a(u), if p ≡ 1 (mod 2q)

Lemma 2.1 The sum of all the elements a(i)′s in one period p equals to zero [3]. In

other words,p−1∑x=0

a(x) = 0 (2.2)

proof: Since every Ci has T elements, the left summands of (2.2) can be divided as

follows.p−1∑x=1

a(x) =∑x∈C0

a(x) +∑x∈C1

a(x) + ... +∑

x∈Cq−1

a(x)

= T · ej 2π·0q + T · ej 2π·1

q + ... + T · ej2π·(q−1)

q

= T · (1 + ej 2π

q + ej 4π

q + ... + ej

2π(q−1)q )

= T · (1 − ej 2π·q

q

1 − ej 2π

q

) = 0.

Theorem 2.1 If a(u) = α(u) + jβ(u) (where α(u) and β(u) are real part and imag-

inary part of a(u) respectively.), then the autocorrelation of q-phase PRS is given as

follows. [3]

Ra(τ) =p−1∑x=0

a(x)a(x + τ)∗ =

{−1 − j · 2β(τ), if p ≡ q + 1 mod 2q

−1 + 2α(τ), if p ≡ 1 mod 2q

for any τ �= 0 (mod p)

8

proof: We assume that τ �= 0 (mod p). Then,

Ra(τ) =p−1∑x=0

a(x)a(x + τ)∗

= [p−1∑x=0

a(x)∗a(x + τ)]∗ = [p−1∑x=0

a(x + τ)a(x)∗]∗

= [a(τ)a(0)∗ + a(0)a(p − τ)∗ +p−1∑

x=1,x �=p−τ

a(x + τ)a(x)∗]∗

= [a(τ) + a(−τ)∗ +p−1∑

x=1,x �=p−τ

a(x + τ

x)]∗

= a(τ)∗ + a(−τ) + [p−1∑

x=1,x �=p−τ

a(1 + τ · x−1)]∗. (2.3)

We claim that

{a(1 + τ · x−1) | 1 ≤ x ≤ p − 1, x �= p − τ} = {a(j) | 2 ≤ j ≤ p − 1}.

Define, for a given τ ∈ Zp∗ � Zp \ {0}

ϕτ : Zp∗ −→ Zp

∗

x �→ ϕτ (x) = τ · x−1

where Zp∗ is the set of nonzero integers modulo p.

Clearly ϕτ is a well-defined map. Furthermore ϕτ is bijective, since ϕτ (x1) = ϕτ (x2)

implies τ · x−11 = τ · x−1

2 or x1 = x2. Thus{τ · x−1 | x ∈ Zp

∗} ={τ · x−1 | 1 ≤ x ≤ p − 1} = {1 ≤ x ≤ p − 1

}= Zp

∗.

S �{a(τ · x−1) | 1 ≤ x ≤ p − 1

}= {a(1), a(2), . . . , a(p − 2), a(p − 1)}

⇔ {a(1 + τ · x−1) | a(2), a(3), . . . , a(p − 1), a(0)} = S \ {a(1)} ∪ {a(0)}.

9

Observe that if x = p − τ , then

a(1 + τ · x−1) = a(1 + τ · (p − τ)−1) = a(1 + τ · (−τ)−1) = a(1 − τ · τ−1) = a(1−1) = a(0).

Hence,

{a(1 + τ · x−1) | 1 ≤ x ≤ p − 1, x �= p − τ} = S \ {a(1)} � S̃

Finally,

p−1∑x=1,x �=p−τ

a(1 + τ · x−1) =∑s∈S̃

s =∑s∈S

s − a(1) =p−1∑j=1

a(j) − a(1) = 0 − a(1) = −a(1) = −1.

Therefore (2.3) becomes as follows.

Ra(τ) = a(τ)∗ + a(−τ) + [p−1∑

x=1,x �=p−τ

a(1 + τ · x−1)]∗

= a(τ)∗ + a(−τ) + (−1).

We know that a(τ) = α(τ) + j · β(τ) implies a(τ)∗ = α(τ) − j · β(τ).

Now, when p ≡ q + 1 (mod 2q), a(−τ) = −a(τ) = −α(τ) − j · β(τ). Thus,

Ra(τ) = a(τ)∗ + a(−τ) + (−1) = −1 − j · 2β(τ).

When p ≡ 1 (mod 2q), a(−τ) = a(τ) = α(τ) + j · β(τ). Thus,

Ra(τ) = a(τ)∗ + a(−τ) + (−1) = −1 + 2α(τ).

From Theorem 2.1 the absolute value of autocorrelation function of q-phase PRS of

length p is equal to or less than 3. In other words |Ra(τ)| ≤ 3. And it is always true

regardless of sequence length p and the number of phase q.

10

Chapter 3

Crosscorrelation of PolyphasePower Residue Sequences

3.1 Set of distinct q-phase Power Residue Sequence

We reviewed a well-known construction of PPRS in Chapter 2. We know that the role

of a primitive root µ is to partition the nonzero integers into cosets Ci. In general, there

are φ(p− 1) primitive roots in Zp. If we use another primitive root µ′ to make a q-phase

PRS, what would happen? It may be (not always) different from the original sequence.

In Example 2.3, we constructed a 3-phase PRS. There exist φ(12) = 4 primitive roots in

Z13. These are 21 = 2, 25 = 6, 27 = 11, 211 = 7. Following example shows how the

PPRS changes if we change the primitive root in the construction described in Chapter

2.

Example 3.5 p=13, q=3

i 1 2 3 4 5 6 7 8 9 10 11 12

i3(mod 13) 1 8 1 12 8 8 5 5 1 12 5 12

11

i 0 1 2 3 4 5 6 7 8 9 10 11

2i(mod 13) 1 2 4 8 3 6 12 11 9 5 10 7

6i(mod 13) 1 6 10 8 9 2 12 7 3 5 4 11

11i(mod 13) 1 11 4 5 3 7 12 2 9 8 10 6

7i(mod 13) 1 7 10 5 9 11 12 6 3 8 4 2

When µ=2,

C0 = {1, 5, 8, 12}

C1 = µ1 · C0 = 2 · C0 = 2 · {1, 5, 8, 12} = {2, 10, 3, 11}

C2 = µ2 · C0 = 22 · C0 = 4 · {1, 5, 8, 12} = {4, 7, 6, 9}

When µ=6,

C0 = {1, 5, 8, 12}

C1 = µ1 · C0 = 6 · C0 = 6 · {1, 5, 8, 12} = {6, 4, 9, 7}

C2 = µ2 · C0 = 62 · C0 = 10 · {1, 5, 8, 12} = {10, 11, 2, 3}

When µ=11,

C0 = {1, 5, 8, 12}

C1 = µ1 · C0 = 11 · C0 = 11 · {1, 5, 8, 12} = {11, 3, 10, 2}

C2 = µ2 · C0 = 112 · C0 = 4 · {1, 5, 8, 12} = {4, 7, 6, 9}

When µ=7,

C0 = {1, 5, 8, 12}

C1 = µ1 · C0 = 7 · C0 = 7 · {1, 5, 8, 12} = {7, 9, 4, 6}

C2 = µ2 · C0 = 72 · C0 = 10 · {1, 5, 8, 12} = {10, 11, 2, 3}

12

So 3-ary PRS s(n) of length 13 using µ = 2, 6, 11, 7 are given as follows.

µ s(0) s(1) s(2) s(3) s(4) s(5) s(6) s(7) s(8) s(9) s(10) s(11) s(12)

2 0 0 1 1 2 0 2 2 0 2 1 1 0

6 0 0 2 2 1 0 1 1 0 1 2 2 0

11 0 0 1 1 2 0 2 2 0 2 1 1 0

7 0 0 2 2 1 0 1 1 0 1 2 2 0

In Example 3.5, when µ = 2 and µ = 11, the resulting PRS is the same. Likewise µ = 6

and µ = 7 give the same PRS. In general, we may ask the following question: what is

a sufficient condition for two distinct primitive roots mod p to generate the same q-ary

PRS? The next interesting question would be the following: how are two distinct q-

ary PRS of length p from two different primitive roots mod p related? We recall that

the set of all the q-th power residues C0 is the set of the roots of the form µqk, where

k = 0, 1, . . . , T − 1.

Theorem 3.2 Let µ be a primitive root mod p. Let a q-ary PRS {s(n)} be constructed

using µi and {t(n)} be constructed using µj , where both i and j are relatively prime to

p − 1. Then, there exists a constant v (mod q) such that t(n) ≡ v−1 · s(n) (mod q) for

all n, where v is a solution to j ≡ i · v(mod p − 1).

Proof: Let p − 1 = q · T . Since both i and j are relatively prime to p − 1, there

always exists an integer v such that j ≡ i ·v (mod p−1). Therefore, iv = j +n · (p−1)

for some integer n. We can partition all the nonzero integers mod p into q cosets as

described in Chapter 2. Let Ck be the coset obtained by using µi and let Dk be the coset

obtained by using µj for 0 ≤ k ≤ q − 1. We claim that C0 ≡ D0 (mod p). Since both

13

j and i−1 are relatively prime to p − 1, (v, p − 1) = 1. Hence, (v, qT ) = 1 implies

(v, q) = 1 and (v, T ) = 1.

Since (v, T ) = 1,

{0, 1, 2, . . . , T − 1} ≡ v · {0, 1, 2, . . . , T − 1} (mod T )

q · {0, 1, 2, . . . , T − 1} ≡ qv · {0, 1, 2, . . . , T − 1} (mod qT )

Therefore,

C0 ≡{µ0, µiq, µ2iq, . . . , µi(T−1)q} ≡ {µ0, µiqv, µ2iqv, . . . , µi(T−1)qv}

≡{µ0, µq(j+n(p−1)), µ2q(j+n(p−1)), . . . , µ(T−1)q(j+n(p−1))}

≡{µ0, µjq, µ2jq, . . . , µj(T−1)q} ≡ D0 (mod p)

We claim that Dk ≡ Cvk (mod p) for all 0 ≤ k ≤ T − 1,

Dk ≡ (µj)k · D0 ≡ (µiv+m(p−1))k · C0

≡ µivk+mk(p−1) · C0 ≡ µivk · C0 (3.1)

≡ (µi)vk · C0 ≡ Cvk (mod p)

We use the fact that µmk(p−1) = µmkqT ∈ C0 means µmk(p−1) · C0 = C0 in (3.1).

t(n) = k ⇔ n ∈ Dk ≡ Cvk (mod p) ⇔ s(n) = vk (mod q)

Hence, s(n) ≡ v · t(n) (mod q) for all n. It completes the proof.

�

Corollary 3.1 Let µ be a primitive root mod p. Let a q-ary PRS {s(n)} be constructed

using µi and {t(n)} be constructed using µj , where both i and j are relatively prime to

14

p − 1. s(n) ≡ t(n) (mod q) for all n if and only if v ≡ 1 (mod q) if and only if i ≡ j

(mod q).

Proof: From Theorem 3.2, clearly s(n) ≡ t(n) (mod q) ⇔ i ≡ j (mod q).

Assume that v ≡ 1 (mod q). Then v = qm + 1 for some integer m. Hence,

j ≡ i · (1 + q · m) (mod p − 1)

≡ i + iqm (mod p − 1)

⇔ j ≡ i (mod q)

Conversely, assume that i ≡ j (mod q).

v ≡ j

i(mod p − 1)

≡ qm + i

i(mod p − 1)

≡ q · m

i+ 1(mod p − 1)

⇔ v ≡ 1(mod p − 1)

�

In Example 3.5, when µ = 2 and µ = 11, the resulting 3-ary PRS is the same. Since

21 = 2, 27 = 11 and 1 ≡ 7 (mod 3), it coincides with Corollary 3.1. Likewise we

can check easily in the cases of µ = 25 = 6 and µ = 211 = 7. In Example 3.5, for a

3-ary PRS {s1(n)} using 21 = 2 or 27 = 11, another 3-ary PRS {s2(n)} using 25 = 6

or 211 = 7 is related as s2(n) = 2 · s1(n) mod 3. It is because 5 ≡ 11 ≡ 2 mod 3 and

2−1 = 2 mod 3. Hence, for a given q-ary PRS {s1(n)}, another q-ary PRS {s2(n)} is

related as multiplying an integer t to s1(n), where (t, q) = 1.

15

Then can all the other q-ary PRS’s be constructed by multiplying an integer t satisfying

(t, q) = 1 to the original sequence instead of changing a primitive root? Following

Theorem will be able to give us the solution.

Theorem 3.3 Let p be an odd prime and q be a divisor of p − 1. Then,

Up−1 ≡ Uq (mod q).

where Up−1 and Uq are the unit groups of Zp−1 and Zq, respectively.

Proof: If an integer t satisfies (t, p − 1) = 1, then (t, q) = 1 for a divisor q of p − 1.

Hence U ′p−1 ⊂ Uq. We only have to show that Uq − U ′

p−1 is an empty set. If not, there

exist some r ∈ Uq such that

(r + kq, p − 1) > 1 (3.2)

for all k = 0, 1, · · · , T − 1.

Here, (3.2) implies (r + kq, T ) > 1 for all k = 0, 1, · · · , T − 1 since r ∈ Uq implies

that (r + kq, q) = 1. Now, let d = (q, T )

When d = 1,

R = {r + kq (mod T ) | 0 ≤ k ≤ T − 1} = {0, 1, 2, · · · , T − 1}

Thus, there are φ(T ) elements in R which is relatively prime to T . It contradicts to the

assumption.

When d > 1,

The set R can be written as

R = {r, r + d, r + 2d, · · · , r + (T

d− 1)d}

16

The elements in R are already relatively prime to q, thus relatively prime to d. Therefore,

finding an element in R that is relatively prime to T becomes finding an element that

is relatively prime to Td . In other words, the process goes to the first stage, with the

replacements q → d, and T → Td . When we do these procedure repeatedly, then finally,

we reach the moments that (d, Td ) = 1.

�

Corollary 3.2 The number of all the distinct q-ary PRS is exactly φ(q).

Proof: Obvious.

In Table 3.1, we present all the distinct q-ary PRS for a given length p and a phase

q. For a given length p, all the divisors of p − 1 except for 1, 2, p − 1 are taken into

consideration as q. When q is 1 or p−1, the sequence becomes more or less trivial. When

q is 2, the sequence becomes the binary Legendre sequence. Here the column ’exponents

of µ’ contains the set of exponents of primitive roots that lead to the same PRS. Of course

the exponents are all relatively prime to p − 1 for primitivity. For instance, if p=13,

µ=2, q=3 is the case, we can obtain a sequence using the primitive roots 21 and 27 and

another sequence using 25 and 211. For convenience, we use q-ary PRS {s(n)} instead

of complex q-phase PRS {a(n)}. Recall that a(n) = ws(n), where w is a complex

primitive q-th root of unity or w = exp(j 2πq ) and s(n) takes on the values in Zq.

In Table 3.2 and Table 3.3 we present only primitive pairs giving the same PPRS for the

length p ≤ 73. These data show that all the exponents of the primitive roots giving the

same PPRS are congruent mod q to each other.

17

Table 3.1: All the q-ary PRS of length p up to 23p µ q exponents of µ q-ary PRS11 2 5 1 0 0 1 3 2 4 4 2 3 1 0

3 0 0 2 1 4 3 3 4 1 2 07 0 0 3 4 1 2 2 1 4 3 09 0 0 4 2 3 1 1 3 2 4 0

13 2 3 1,7 0 0 1 1 2 0 2 2 0 2 1 1 05,11 0 0 2 2 1 0 1 1 0 1 2 2 0

4 1,5 0 0 1 0 2 1 1 3 3 0 2 3 27,11 0 0 3 0 2 3 3 1 1 0 2 1 2

6 1,7 0 0 1 4 2 3 5 5 3 2 4 1 05,11 0 0 5 2 4 3 1 1 3 4 2 5 0

17 3 4 1,5,9,13 0 0 2 1 0 1 3 3 2 2 3 3 1 0 1 2 03,7,11,15 0 0 2 3 0 3 1 1 2 2 1 1 3 0 3 2 0

8 1,9 0 0 6 1 4 5 7 3 2 2 3 7 5 4 1 6 03,11 0 0 2 3 4 7 5 1 6 6 1 5 7 4 3 2 05,13 0 0 6 5 4 1 3 7 2 2 7 3 1 4 5 6 07,15 0 0 2 7 4 3 1 5 6 6 5 1 3 4 7 2 0

19 2 3 1,7,3 0 0 1 1 2 1 2 0 0 2 2 0 0 2 1 2 1 1 05,11,17 0 0 2 2 1 2 1 0 0 1 1 0 0 1 2 1 2 2 0

6 1,7,13 0 0 1 1 2 4 2 0 3 2 5 0 3 5 1 5 4 4 35,11,17 0 0 5 5 4 2 4 0 3 4 1 0 3 1 5 1 2 2 3

9 1 0 0 1 4 2 7 5 6 3 8 8 3 6 5 7 2 4 1 05 0 0 2 8 4 5 1 3 6 7 7 6 3 1 5 4 8 2 07 0 0 4 7 8 1 2 6 3 5 5 3 6 2 1 8 7 4 011 0 0 5 2 1 8 7 3 6 4 4 6 3 7 8 1 2 5 013 0 0 7 1 5 4 8 6 3 2 2 3 6 8 4 5 1 7 017 0 0 8 5 7 2 4 3 6 1 1 6 3 4 2 7 5 8 0

23 5 11 1 0 0 2 5 4 1 7 8 6 10 3 9 9 3 10 6 8 7 1 4 5 2 03 0 0 8 9 5 4 6 10 2 7 1 3 3 1 7 2 10 6 4 5 9 8 05 0 0 7 1 3 9 8 6 10 2 5 4 4 5 2 10 6 8 9 3 1 7 07 0 0 5 7 10 8 1 9 4 3 2 6 6 2 3 4 9 1 8 10 7 5 09 0 0 10 3 9 5 2 7 8 6 4 1 1 4 6 8 7 2 5 9 3 10 013 0 0 1 8 2 6 9 4 3 5 7 10 10 7 5 3 4 9 6 2 8 1 015 0 0 6 4 1 3 10 2 7 8 9 5 5 9 8 7 2 10 3 1 4 6 017 0 0 4 10 8 2 3 5 1 9 6 7 7 6 9 1 5 3 2 8 10 4 019 0 0 3 2 6 7 5 1 9 4 10 8 8 10 4 9 1 5 7 6 2 3 021 0 0 9 6 7 10 4 3 5 1 8 2 2 8 1 5 3 4 10 7 6 9 0

18

Table 3.2: Sets of exponents of µ that gives the same q-ary PRS of length p ≤ 73.p µ q exponents of µ

29 2 4 {1,5,9,13,17,25}, {3,11,15,19,23,27}7 {1,15}, {3,17}, {5,19}, {9,23}, {11,25}, {13,27}14 {1,15}, {3,17}, {5,19}, {9,23}, {11,25}, {13,27}

31 3 3 {1,7,13,19}, {11,17,23,29}5 {1,11}, {7,17}, {13,23}, {19,29}6 {1,7,13,19}, {11,17,23,29}10 {1,11}, {7,17}, {13,23}, {19,29}15 {1}, {7}, {11}, {13}, {17}, {19}, {23}, {29}

37 2 3 {1,7,13,19,25,31}, {5,11,17,23,29,35}4 {1,5,13,17,25,29}, {7,11,19,23,31,35}6 {1,7,13,19,25,31}, {5,11,17,23,29,35}9 {1,19}, {5,23}, {7,25}, {11,29}, {13,31}, {17,35}12 {1,13,25}, {5,17,29}, {7,19,31}, {11,23,35}18 {1,19}, {5,23}, {7,25}, {11,29}, {13,31}, {17,35}

41 6 4 {1,9,13,17,21,29,33,37}, {3,7,11,19,23,27,31,39}5 {1,11,21,31}, {3,13,23,33}, {7,17,27,37,}, {9,19,29,39}8 {1,9,17,33}, {3,11,19,27}, {7,23,31,39}, {13,21,29,37}10 {1,11,21,31}, {3,13,23,33}, {7,17,27,37,}, {9,19,29,39}20 {1,21}, {3,23}, {7,27}, {9,29}, {11,31}, {13,33}, {17,37}, {19,39}

43 3 3 {1,13,19,25,31,37}, {5,11,17,23,29,41}6 {1,13,19,25,31,37}, {5,11,17,23,29,41}7 {1,29}, {5,19}, {11,25}, {13,41}, {17,31}, {23,37}14 {1,29}, {5,19}, {11,25}, {13,41}, {17,31}, {23,37}21 {1}, {5}, {11}, {13}, {17}, {19}, {23}, {25}, {29}, {31}, {37}, {41}

47 5 23 {1},{3},{5},{7},{9},{11},{13},{15},{17},{19},{21},{25},{27},{29},{31},{33},{35},{37},{39},{41},{43},{45}

53 2 4 {1,5,9,17,21,25,29,33,37,41,45,49},{3,7,11,15,19,23,27,31,35,43,47,51}13 {1,27},{3,29},{5,31},{7,33},{9,35},{11,37},

{15,41},{17,43},{19,45},{21,47},{23,49},{25,51}26 {1,27},{3,29},{5,31},{7,33},{9,35},{11,37},

{15,41},{17,43},{19,45},{21,47},{23,49},{25,51}59 2 29 {1},{3},{5},{7},{9},{11},{13},{15},{17},{19},{21},{23},{25},{27},{31},

{33},{35},{37},{39},{41},{43},{45},{47},{49},{51},{53},{55},{57}61 2 3 {1,7,13,19,31,37,43,49},{11,17,23,29,41,47,53,59}

4 {1,13,17,29,37,41,49,53},{7,11,19,23,31,43,47,59}5 {1,11,31,41},{7,17,37,47},{13,23,43,53},{19,29,49,59}6 {1,7,13,19,31,37,43,49},{11,17,23,29,41,47,53,59}

19

Table 3.3: Sets of exponents of µ that gives the same q-ary PRS of length p ≤ 73(continued).

p µ q exponents of µ

61 2 10 {1,11,31,41},{7,17,37,47},{13,23,43,53},{19,29,49,59}12 {1,13,37,49},{7,19,31,43},{11,23,47,59},{17,29,41,53}15 {1,31},{7,37},{11,41},{13,43},{17,47},{19,49},{23,53},{29,59}20 {1,41},{7,47},{11,31},{13,53},{17,37},{19,59},{23,43},{29,49}30 {1,31},{7,37},{11,41},{13,43},{17,47},{19,49},{23,53},{29,59}

67 2 3 {1,7,13,19,25,31,37,43,49,61},{5,17,23,29,35,41,47,53,59,65}6 {1,7,13,19,25,31,37,43,49,61},{5,17,23,29,35,41,47,53,59,65}11 {1,23},{5,49},{7,29},{13,35},{17,61},{19,41},{25,47},{31,53},{37,59},{43,65}22 {1,23},{5,49},{7,29},{13,35},{17,61},{19,41},{25,47},{31,53},{37,59},{43,65}33 {1},{5},{7},{13},{17},{19},{23},{25},{29},{31},{35},

{37},{41},{43},{47},{49},{53},{59},{61},{65}71 7 5 {1,11,31,41,51,61},{3,13,23,33,43,53},

{9,19,29,39,59,69},{17,27,37,47,57,67}7 {1,29,43,57},{3,17,31,59},{9,23,37,51},

{11,39,53,67},{13,27,41,69},{19,33,47,61}10 {1,11,31,41,51,61},{3,13,23,33,43,53},

{9,19,29,39,59,69},{17,27,37,47,57,67}14 {1,29,43,57},{3,17,31,59},{9,23,37,51},

{11,39,53,67},{13,27,41,69},{19,33,47,61}35 {1},{3},{9},{11},{13},{17},{19},{23},{27},{29},{31},{33},

{37},{39},{41},{43},{47},{51},{53},{57},{59},{61},{67},{69}73 5 3 {1,7,13,19,25,31,37,43,49,55,61,67},

{5,11,17,23,29,35,41,47,53,59,65,71}4 {1,5,13,17,25,29,37,41,49,53,61,65},

{7,11,19,23,31,35,43,47,55,59,67,71}6 {1,7,13,19,25,31,37,43,49,55,61,67},

{5,11,17,23,29,35,41,47,53,59,65,71}8 {1,17,25,41,49,65},{5,13,29,37,53,61},

{7,23,31,47,55,71},{11,19,35,43,59,67}9 {1,19,37,55},{5,23,41,59},{7,25,43,61},

{11,29,47,65},{13,31,49,67},{17,35,53,71}12 {1,13,25,37,49,61},{5,17,29,41,53,65},

{7,19,31,43,55,67},{11,23,35,47,59,71}18 {1,19,37,55},{5,23,41,59},{7,25,43,61},

{11,29,47,65},{13,31,49,67},{17,35,53,71}

20

3.2 Crosscorrelation of two q-phase PRS of length p

The periodic crosscorrelation between two q-phase sequences a(n) and b(n)(of period

p) is defined by

Ca,b(τ) =p−1∑n=0

a(n)b(n + τ)∗

=p−1∑n=0

ws1(n)−s2(n+τ) (3.3)

where w is a complex q-th root of unity and a(n) = ws1(n) and b(n) = ws2(n).

Now we will discuss the crosscorrelation of PPRS. For convenience we will consider

the absolute value of (3.3) as the crosscorrelation of PPRS. In Tables 3.4 and 5.1,

we compute the crosscorrelation between two distinct q-phase PRS’s of the same length.

Table 5.1 is in Appendix. In these Tables, (i, j) denotes the computation of the cross-

correlation of PRS’s made from µi and µj . Since there are φ(q) distinct PRS, we con-

sider the crosscorrelation for all the(φ(q)

2

)PRS pairs. Here ’max

τC(τ)’ is the maximum

crosscorrelation for all the time shift τ . From Tables 3.4 and 5.1, we observe that the

difference of ’maxτ

C(τ)’ and√

p is less than 2.

Now we compare the maxall pairs

{maxτ

C(τ)} with√

p + 2. In Tables 3.5, 3.6, and 3.7,

we present maxall pairs

{maxτ

C(τ)} and√

p + 2 for all the prime p ≤ 97. In addition, we

present some imformations about the levels of all the crosscorrelation. For two distinct

PPRS {a(n)} and {b(n)}, let L be the number of levels of crosscorrelation between

them. In Tables 3.5, 3.6, and 3.7, the column ’mLv’ denotes the minimum of L and

’MLv’ denotes the maximum of L. In these tables, we see that no crosscorrelation is

larger than√

p + 2 for all p ≤ 97. So, we suggest the following conjecture.

21

Conjecture 3.1 Let p be an odd prime and q be a divisor of p− 1. The crosscorrelation

of two distinct q-phase PRS a(n) and b(n) of length p is upper-bounded to√

p + 2. i.e.

|Ca,b(τ)| ≤ √p + 2

for all the time shift τ .

22

Table 3.4: Maximum Crosscorrelation of PRS’s of length p ≤ 19.p µ q (i, j) max

τC(τ) p/3

√p

11 2 5 (1,3) 5.213 3.667 3.317(1,7) 4.765(1,9) 5.213(3,7) 5.314(3,9) 4.765(7,9) 5.213

13 2 3 (1,5) 5.568 4.333 3.6064 (1,7) 3.6066 (1,5) 5.568

17 3 4 (1,3) 6.083 5.667 4.1238 (1,3) 6.083

(1,5) 5.745(1,7) 6.123(3,5) 6.123(3,7) 5.745(5,7) 6.083

19 2 3 (1,5) 6.083 6.333 4.3596 (1,5) 4.3599 (1,5) 5.752

(1,7) 5.392(1,11) 6.34(1,13) 6.282(1,17) 6.314(5,7) 6.282(5,11) 5.875(5,13) 6.282(5,17) 6.282(7,11) 6.34(7,13) 5.875(7,17) 6.34(11,13) 6.282(11,17) 5.392(13,17) 5.752

23

Table 3.5: Maximum and the number of levels of Crosscorrelations of PPRS.

p µ q# of distinct

PPRSmax

all pairs{max

τC(τ)} √

p + 2 mLv MLv

11 2 5 4 5.314 5.317 6 6

13 2 3 2 5.568 5.606 4 4

4 2 3.606 2 2

6 2 5.568 7 7

17 3 4 2 6.083 6.123 5 5

8 4 6.123 5 9

19 2 3 2 6.083 6.359 4 4

6 2 4.359 2 2

9 6 6.34 9 10

23 5 11 10 6.796 6.796 12 12

29 2 4 2 5.385 7.385 2 2

7 6 7.364 8 8

14 6 7.38 15 15

31 3 3 2 7 7.568 4 4

5 4 7.562 6 6

6 2 5.568 2 2

10 4 7.434 2 10

15 8 7.568 9 16

37 2 3 2 7.81 8.083 4 4

4 2 6.083 2 2

6 2 7.937 7 7

9 6 8.081 9 10

12 4 8.078 2 8

18 6 8.081 17 19

41 6 4 2 8.062 8.403 5 5

5 4 8.402 6 6

8 4 8.394 2 6

10 4 8.402 11 11

20 8 8.402 12 21

24


p µ q# of distinct

PPRSmax

all pairs{max

τC(τ)} √

p + 2 mLv MLv

43 3 3 2 8.544 8.557 4 4

6 2 6.557 2 2

7 6 8.537 8 8

14 6 8.505 2 14

21 12 8.556 13 22

47 5 23 22 8.856 8.856 24 24

53 2 4 2 7.28 9.28 2 2

13 12 9.28 14 14

26 12 9.28 27 27

59 2 29 28 9.681 9.681 30 30

61 2 3 2 9.539 9.81 4 4

4 2 7.81 2 2

5 4 9.804 6 6

6 2 9.644 7 7

10 4 9.81 11 11

12 4 9.807 2 8

15 8 9.809 10 16

20 8 9.804 2 20

30 8 9.81 18 31

67 2 3 2 10.149 10.185 4 4

6 2 8.185 2 2

11 10 10.185 12 12

22 10 10.162 2 22

33 20 10.185 19 34

71 7 5 4 10.404 10.426 6 6

7 6 10.425 8 8

10 4 10.311 2 10

14 6 10.376 2 14

35 24 10.426 21 36

25


p µ q# of distinct

PPRSmax

all pairs{max

τC(τ)} √

p + 2 mLv MLv

73 5 3 2 9.849 10.544 4 4

4 2 10.44 5 5

6 2 10.536 7 7

8 4 10.533 2 6

9 6 10.542 9 10

12 4 10.536 8 13

18 6 10.542 17 19

24 8 10.538 2 14

36 12 10.544 20 37

79 3 3 2 10.44 10.888 4 4

6 2 8.888 2 2

13 12 10.888 14 14

26 12 10.873 2 26

39 24 10.888 22 40

83 2 41 40 11.11 11.11 42 42

89 3 4 2 11.18 11.434 5 5

8 4 11.392 2 6

11 10 11.433 12 12

22 10 11.433 23 23

44 20 11.434 24 45

97 5 3 2 11.79 11.849 4 4

4 2 11.705 5 5

6 2 11.79 7 7

8 4 11.796 6 9

12 4 11.84 8 13

16 8 11.847 10 17

24 8 11.84 13 25

32 16 11.848 2 30

48 16 11.849 25 49

26

Chapter 4

Transformation of q-ary PRS andtheir crosscorrelations

In general, for a given q-ary sequence s(n) of length p, the following transformations

can be taken into consideration.

(T-i) Constant Multiplying:

s(n) → k · s(n)

where (k, q) = 1.

(T-ii) Affine Shift:

s(n) → n · k1 + k2 + s(n)

where 0 ≤ k1, k2 ≤ q − 1.

(T-iii) Cyclic Shift:

s(n) → s(n + τ)

where τ is an integer and n + τ is computed modulo p.

27

(T-iv) Decimation:

s(n) → s(dn)

where d is an integer satisfying (d, n) = 1 and dn is computed modulo p.

Let M(s(n)), A(s(n)), C(s(n)) and D(s(n)) denote the family of the sequences

obtained by the transformations (T-i), (T-ii), (T-iii) and (T-iv) respectively.

Example 4.6 When p=13 and q=4, a 4-ary sequence is given as follows.

{s(n)} = {0 0 1 0 2 1 1 3 3 0 2 3 2}

1. M(s(n))

n 0 1 2 3 4 5 6 7 8 9 10 11 12

1 · s(n)(mod 4) 0 0 1 0 2 1 1 3 3 0 2 3 23 · s(n)(mod 4) 0 0 3 0 2 3 3 1 1 0 2 1 2

2. C(s(n))

n 0 1 2 3 4 5 6 7 8 9 10 11 12

s(n + 0 mod 13) 0 0 1 0 2 1 1 3 3 0 2 3 2s(n + 1 mod 13) 0 1 0 2 1 1 3 3 0 2 3 2 0s(n + 2 mod 13) 1 0 2 1 1 3 3 0 2 3 2 0 0s(n + 3 mod 13) 0 2 1 1 3 3 0 2 3 2 0 0 1s(n + 4 mod 13) 2 1 1 3 3 0 2 3 2 0 0 1 0s(n + 5 mod 13) 1 1 3 3 0 2 3 2 0 0 1 0 2s(n + 6 mod 13) 1 3 3 0 2 3 2 0 0 1 0 2 1s(n + 7 mod 13) 3 3 0 2 3 2 0 0 1 0 2 1 1s(n + 8 mod 13) 3 0 2 3 2 0 0 1 0 2 1 1 3s(n + 9 mod 13) 0 2 3 2 0 0 1 0 2 1 1 3 3s(n + 10 mod 13) 2 3 2 0 0 1 0 2 1 1 3 3 0s(n + 11 mod 13) 3 2 0 0 1 0 2 1 1 3 3 0 2s(n + 12 mod 13) 2 0 0 1 0 2 1 1 3 3 0 2 3

28

3. A(s(n))

(k1, k2) 0 1 2 3 4 5 6 7 8 9 10 11 12

(0, 0) 0 0 1 0 2 1 1 3 3 0 2 3 2(0, 1) 1 1 2 1 3 2 2 0 0 1 3 0 3(0, 2) 2 2 3 2 0 3 3 1 1 2 0 1 0(0, 3) 3 3 0 3 1 0 0 2 2 3 1 2 1(1, 0) 0 1 3 3 2 2 3 2 3 1 0 2 2(1, 1) 1 2 0 0 3 3 0 3 0 2 1 3 3(1, 2) 2 3 1 1 0 0 1 0 1 3 2 0 0(1, 3) 3 0 2 2 1 1 2 1 2 0 3 1 1(2, 0) 0 2 1 2 2 3 1 1 3 2 2 1 2(2, 1) 1 3 2 3 3 0 2 2 0 3 3 2 3(2, 2) 2 0 3 0 0 1 3 3 1 0 0 3 0(2, 3) 3 1 0 1 1 2 0 0 2 1 1 0 1(3, 0) 0 3 3 1 2 0 3 0 3 3 0 0 2(3, 1) 1 0 0 2 3 1 0 1 0 0 1 1 3(3, 2) 2 1 1 3 0 2 1 2 1 1 2 2 0(3, 3) 3 2 2 0 1 3 2 3 2 2 3 3 1

4. D(s(n))

n 0 1 2 3 4 5 6 7 8 9 10 11 12

s(1 · n mod 13) 0 0 1 0 2 1 1 3 3 0 2 3 2s(2 · n mod 13) 0 1 2 1 3 2 2 0 0 1 3 0 3s(3 · n mod 13) 0 0 1 0 2 1 1 3 3 0 2 3 2s(4 · n mod 13) 0 2 3 2 0 3 3 1 1 2 0 1 0s(5 · n mod 13) 0 1 2 1 3 2 2 0 0 1 3 0 3s(6 · n mod 13) 0 1 2 1 3 2 2 0 0 1 3 0 3s(7 · n mod 13) 0 3 0 3 1 0 0 2 2 3 1 2 1s(8 · n mod 13) 0 3 0 3 1 0 0 2 2 3 1 2 1s(9 · n mod 13) 0 0 1 0 2 1 1 3 3 0 2 3 2s(10 · n mod 13) 0 2 3 2 0 3 3 1 1 2 0 1 0s(11 · n mod 13) 0 3 0 3 1 0 0 2 2 3 1 2 1s(12 · n mod 13) 0 2 3 2 0 3 3 1 1 2 0 1 0

29

Now let us apply these transformations to a q-ary PRS s(n) of prime length p. In fact,

the set of these sequences obtained by the constant multiplying is equivalent to the set

of sequences which are made by changing the primitive roots. So in this case, the size

of family M(s(n)) is φ(q). In Chapter 3, we observe that the crosscorrelation in the

family M(s(n)) is upper bounded to√

p + 2 for a given prime length p ≤ 97. Here,

we consider the affine shift. Since the constant k1 and k2 can be all of the integers

in Zq, we can get q2 sequences from the affine shift. The set of all the sequences by

these two transformation can be regarded as A(M(s(n))). So these transformations

give us φ(q) · q2 sequences. Up to p = 521 we compute the crosscorrelations of all of

the q-ary sequences in A(M(s(n))). In Table 4.1 we present the q-ary sequences pairs

(k, k1, k2), (k′, k′1, k

′2) satisfying the bound

√p + 2. Here k and k′ denote the constants

in the constant multiple. k1, k2 and k′1, k

′2 denote the constants in the affine shift. In

Table 5.2, the pairs giving 2-level crosscorrelation are appeared for p ≤ 97. Table 5.2

is in Appendix. Here, we find that the maximum crosscorrelation of the pairs giving 2-

level crosscorrelation is not√

p+2 but√

p . Hence, we suggest the following conjecture

from all of these results.

Conjecture 4.2 Let s(n) be a q-ary PRS of a prime length p. Let M(s(n)) be the set of

all the q-ary sequences obtained by the constant multiple and A(s(n)) be the set of all

the q-ary sequences by the affine shift of s(n). Then the maximum crosscorrelation of

q-ary sequence giving 2-level crosscorrelation in A(M(s(n))) is√

p.

30

Table 4.1: Crosscorrelation of two distict q-ary sequence inA(M(s(n)))

p q (k, k1, k2), (k′, k′1, k

′2) Max Max/p

√p + 2 level

13 3 (1 0 0),(2 0 0) 5.568 0.428 5.606 4(1 0 0),(2 0 1) 5.568 0.428 5.606 4(1 0 0),(2 0 2) 5.568 0.428 5.606 4(1 0 1),(2 0 0) 5.568 0.428 5.606 4(1 0 1),(2 0 1) 5.568 0.428 5.606 4(1 0 1),(2 0 2) 5.568 0.428 5.606 4(1 0 2),(2 0 0) 5.568 0.428 5.606 4(1 0 2),(2 0 1) 5.568 0.428 5.606 4(1 0 2),(2 0 2) 5.568 0.428 5.606 4(1 1 0),(1 2 0) 5.568 0.428 5.606 7(1 1 0),(1 2 1) 5.568 0.428 5.606 7(1 1 0),(1 2 2) 5.568 0.428 5.606 7(1 1 0),(2 2 0) 5.292 0.407 5.606 6(1 1 0),(2 2 1) 5.292 0.407 5.606 6(1 1 0),(2 2 2) 5.292 0.407 5.606 6(1 1 1),(1 2 0) 5.568 0.428 5.606 7(1 1 1),(1 2 1) 5.568 0.428 5.606 7(1 1 1),(1 2 2) 5.568 0.428 5.606 7(1 1 1),(2 2 0) 5.292 0.407 5.606 6(1 1 1),(2 2 1) 5.292 0.407 5.606 6(1 1 1),(2 2 2) 5.292 0.407 5.606 6(1 1 2),(1 2 0) 5.568 0.428 5.606 7(1 1 2),(1 2 1) 5.568 0.428 5.606 7(1 1 2),(1 2 2) 5.568 0.428 5.606 7(1 1 2),(2 2 0) 5.292 0.407 5.606 6(1 1 2),(2 2 1) 5.292 0.407 5.606 6(1 1 2),(2 2 2) 5.292 0.407 5.606 6(1 2 0),(2 1 0) 5.568 0.428 5.606 5(1 2 0),(2 1 1) 5.568 0.428 5.606 5(1 2 0),(2 1 2) 5.568 0.428 5.606 5(1 2 1),(2 1 0) 5.568 0.428 5.606 5(1 2 1),(2 1 1) 5.568 0.428 5.606 5(1 2 1),(2 1 2) 5.568 0.428 5.606 5

31

Now, we consider the decimation. For a given q-ary PRS s(n) of length p, we make

all the sequences by the decimation. The set of all the sequences from decimation is

D(s(n)). Since the length p is a prime, there exist φ(p) = p − 1 integers relatively

prime to p. So there are p − 1 sequences in D(s(n)). In Table 4.2, we compute all

of the crosscorrelations of the 5-ary sequences in D(s(n)), where s(n) is a 5-ary PRS

of length 11. Here, the column ’d1, d2’ denotes the computation of the crosscorrelation

of the sequences from d1-decimation and d2-decimation respectively. For p ≤ 97, we

generate all the sequences from the decimation and compute the crosscorrelations of

them. From the computing, we have known that the crosscorrelation values are much

larger than√

p + 2 and what is worse, they are equal to or larger than p − 2. Recall

that the crosscorrrelation values are upper bounded to√

p + 2 in the case of the con-

stant multiple transformation. In this time, we make D(M(s(n))), the set of all the

sequences by two transformations the decimations and the constant multiple of s(n).

In Table 5.3, we present all of the crosscorrelations of the sequences in D(M(s(n)))

of the length p ≤ 97. Table 5.3 is in the Appendix. The columns k and k′ denote

the constant in the constant multiple. Let s1(n) and s2(n) be the sequences obtained

by multiplying the constant k and k′ to the original q-ary PRS s(n) respectively. Let

D(s1(n)) and D(s2(n)) be the sets of all the sequences considering every d-decimation

of s1(n) and s2(n) respectively, where 1 ≤ d ≤ p − 1. Let a(n) be the d1-decimation

of s1(n) and b(n) be the d2-decimation of s2(n). Now, we compute Maxτ

(Ca,b(τ)) the

maximum crosscorrelation of the sequences a(n) and b(n). In Table 5.3, the column

maxd1,d2

Maxτ

(Ca,b(τ)) denotes the maximum of Maxτ

(Ca,b(τ)) for all 1 ≤ d1, d2 ≤ p − 1.

32

Likewise, the column mind1,d2

Maxτ

(Ca,b(τ)) denotes the minimum of Maxτ

(Ca,b(τ)) for all

1 ≤ d1, d2 ≤ p − 1. We can observe the following. For a given length p and a phase

q, if the sequences a(n) and b(n) are d-decimation and d′-decimation of the same q-ary

PRS(k = k′), then their crosscorrelation is much larger than√

p + 2. On the other hand,

if these sequences a(n) and b(n) are d-decimation and d′-decimation of the distinct q-

ary PRS(k �= k′), then their crosscorrelation is bounded to√

p + 2. Hence, we suggest

the following Conjecture.

Conjecture 4.3 Let p be an odd prime and q be a divisor of p − 1. Let t be an integer

satisfying 1 < t < q and (t, q) = 1. Let s1(n) and s2(n) be two distinct q-ary PRS

such that s1(n) = t · s2(n). Let D(s1(n)) and D(s2(n)) be the set of all the q-ary

sequences obtained by the decimation of s1(n) and s2(n) respectively. For two distinct

q-ary sequences a(n) ∈ D(s1(n)) and b(n) ∈ D(s2(n)), the crosscorrelation of them is

upper bounde to√

p + 2. i.e.

| Ca,b(τ) | ≤ √p + 2.

�

Since s1(n) is an element of D(s1(n)) and s2(n) is an element of D(s2(n)), the

Conjecture 4.3 is the generalized version of the Conjecture 3.1.

33

Table 4.2: Maximum Crosscorrelation of decimations of 5-ary PRS of length 11p q d1, d2 Max

τC(τ)

√p + 2 level

11 5 1,2 10.353 5.317 411 5 1,3 9.210 5.317 411 5 1,4 9.210 5.317 411 5 1,5 10.353 5.317 411 5 1,6 10.353 5.317 411 5 1,7 9.210 5.317 411 5 1,8 9.210 5.317 411 5 1,9 10.353 5.317 411 5 1,10 11.000 5.317 411 5 2,3 9.210 5.317 411 5 2,4 10.353 5.317 411 5 2,5 9.210 5.317 411 5 2,6 9.210 5.317 411 5 2,7 10.353 5.317 411 5 2,8 9.210 5.317 411 5 2,9 11.000 5.317 411 5 2,10 10.353 5.317 411 5 3,4 10.353 5.317 411 5 3,5 10.353 5.317 411 5 3,6 10.353 5.317 411 5 3,7 10.353 5.317 411 5 3,8 11.000 5.317 411 5 3,9 9.210 5.317 411 5 3,10 9.210 5.317 411 5 4,5 9.210 5.317 411 5 4,6 9.210 5.317 411 5 4,7 11.000 5.317 411 5 4,8 10.353 5.317 411 5 4,9 10.353 5.317 411 5 4,10 9.210 5.317 411 5 5,6 11.000 5.317 411 5 5,7 9.210 5.317 411 5 5,8 10.353 5.317 411 5 5,9 9.210 5.317 411 5 5,10 10.353 5.317 411 5 6,7 9.210 5.317 411 5 6,8 10.353 5.317 411 5 6,9 9.210 5.317 411 5 6,10 10.353 5.317 411 5 7,8 10.353 5.317 411 5 7,9 10.353 5.317 411 5 7,10 9.210 5.317 411 5 8,9 9.210 5.317 411 5 8,10 9.210 5.317 411 5 9,10 10.353 5.317 4

34

Chapter 5

Concluding Remarks

As binary Legendre sequences have an ideal autocorrelation, polyphase power residue

sequences have a good autocorrelation property. Regardless of prime length p and phase

q, the absolute values of autocorrelation are upper bounded to 3. Generally we need to

find the sequence family such that have good autocorrelation as well as good crosscorre-

lation for distinguishability of many users in telecommunication system. In this thesis,

we generated the polyphase power residue sequences by changing primitive roots. First

of all we searched all distinct PPRS’s and collect the primitive roots giving the same

PPRS. Then we find the condition to be the same PRS. For two distinct primitive roots

µi and µj , i ≡ j mod q means that the resulting q-ary PRS are the same and vice versa.

Furthermore, we show that for a given q-ary PRS {s(n)}, multiplying an integer t sat-

isfying (t, q) = 1 to s(n) gives us another q-ary PRS. For every integer t such that

(t, q) = 1, there exists an integer t′ ∈ Zp−1 satisfying (t′, p − 1) = 1 and t′ = t

mod q. Hence we prove that there exist φ(q) PPRS. We investigate the crosscorrelation

values between two distinct PPRS’s. The Maximum value of crosscorrelation, compar-

ison with two thresholds(p/3,√

p), etc. are given in Tables. Next, we consider some

35

transformations. The first transformation is a constant multiplication to original PPRS.

Transformation through the constant multiplication gives us same PPRS family with

changing the primitive root. We compute the crosscorrelation of the sequences obtained

by the constant multiple. From the computing, we conjectured that the crosscorrelation

of PPRS is upper-bounded to√

p + 2. The second transformation we consider is the

affine shift. We make the set A(M(s(n))) of the sequences by the former two trans-

formation. Next we compute the crosscorrelations of the sequences in A(M(s(n))).

From the computing, we conjecture that the maximum crosscorrelation of sequences of

PPRS family which have 2-level crosscorrelation is√

p. Finally we consider the deci-

mation. Like the former cases, we make the set D(M(s(n))) of all the sequences from

the decimation and the constant multiple. Then we compute the crosscorrelation of the

sequences in D(M(s(n))). When we apply only the decimation to a q-ary PRS as the

transform, the crosscorrelation is much larger than√

p and what is worse, the crosscor-

relation is equal to or larger than p − 2. On the other hand, when we consider both the

constant multiple and the decimation, an interesting point is observed. If two sequences

are from the distinct q-ary PRS, then the crosscorrelation is upper bounded to√

p + 2

regardless of the decimation. This is suggested as another conjecture. We did not prove

these conjectures algebraically yet. Proof of these will be the next topics.

36

Appendix

Table 5.1: Maximum Crosscorrelation of PRS’s of length 23 ≤ p ≤ 53.

p µ q (i, j) maxτ

C(τ) p/3,√

p p µ q (i, j) maxτ

C(τ) p/3,√

p

23 5 11 (1,3) 6.451 7.667 , 4.796 (9,17) 6.403(1,5) 6.715 (9,19) 6.476(1,7) 6.533 (9,21) 6.217(1,9) 6.794 (13,15) 6.689(1,13) 6.217 (13,17) 6.68(1,15) 6.477 (13,19) 6.794(1,17) 6.61 (13,21) 6.794(1,19) 6.794 (15,17) 6.42(1,21) 6.788 (15,19) 6.715(3,5) 6.633 (15,21) 6.533(3,7) 6.715 (17,19) 6.633(3,9) 6.794 (17,21) 6.715(3,13) 6.476 (19,21) 6.451(3,15) 6.407 29 2 4 (1,3) 5.385 9.667 , 5.385(3,17) 6.407 7 (1,3) 7.074(3,19) 6.744 (1,5) 7.074(3,21) 6.794 (1,9) 6.755(5,7) 6.42 (1,11) 6.789(5,9) 6.68 (1,13) 7.307(5,13) 6.403 (3,5) 7.364(5,15) 6.569 (3,9) 7.074(5,17) 6.773 (3,11) 7.364(5,19) 6.407 (3,13) 6.789(5,21) 6.61 (5,9) 7.291(7,9) 6.689 (5,11) 7.074(7,13) 6.459 (5,13) 6.755(7,15) 6.79 (9,11) 7.364(7,17) 6.569 (9,13) 7.074(7,19) 6.407 (11,13) 7.074(7,21) 6.477 14 (1,3) 7.109(9,13) 6.796 (1,5) 7.307(9,15) 6.459 (1,9) 7.291

continued on nextpage

37

continued from previous pagep µ q (i, j) max

τC(τ) p/3,

√p p µ q (i, j) max

τC(τ) p/3,

√p

(1,11) 7.151 (13,23) 7.511(1,13) 7.38 (13,29) 7.568(3,5) 7.135 (17,19) 7.522(3,9) 6.995 (17,23) 7.562(3,11) 7.375 (17,29) 7.233(3,13) 7.151 (19,23) 7.298(5,9) 7.364 (19,29) 7.511(5,11) 6.995 (23,29) 7.173(5,13) 7.291 37 2 3 (1,5) 7.81 12.333 , 6.083(9,11) 7.135 4 (1,7) 6.083(9,13) 7.307 6 (1,5) 7.937(11,13) 7.109 9 (1,5) 8.081

31 3 3 (1,11) 7 10.333 , 5.568 (1,7) 7.6135 (1,7) 7.562 (1,11) 7.581

(1,13) 7.127 (1,13) 7.57(1,19) 7.492 (1,17) 8.081(7,13) 7.562 (5,7) 8.028(7,19) 7.127 (5,11) 7.927(13,19) 7.562 (5,13) 8.028

6 (1,11) 5.568 (5,17) 7.5710 (1,7) 7.434 (7,11) 8.07

(1,13) 7.187 (7,13) 7.927(1,19) 5.568 (7,17) 7.581(7,13) 5.568 (11,13) 8.028(7,19) 7.187 (11,17) 7.613(13,19) 7.434 (13,17) 8.081

15 (1,7) 7.173 12 (1,5) 8.078(1,11) 7.511 (1,7) 7.796(1,13) 7.233 (1,11) 6.083(1,17) 7.568 (5,7) 6.083(1,19) 7.492 (5,11) 7.796(1,23) 7.238 (7,11) 8.078(1,29) 7.548 18 (1,5) 8.015(7,11) 7.298 (1,7) 7.91(7,13) 7.562 (1,11) 7.912(7,17) 7.511 (1,13) 7.927(7,19) 6.779 (1,17) 8.07(7,23) 7.541 (5,7) 7.946(7,29) 7.238 (5,11) 7.96(11,13) 7.522 (5,13) 8.081(11,17) 7.51 (5,17) 7.927(11,19) 7.547 (7,11) 8.078(11,23) 6.779 (7,13) 7.96(11,29) 7.492 (7,17) 7.912(13,17) 7.568 (11,13) 7.946(13,19) 7.51 (11,17) 7.91


38


τC(τ) p/3,


τC(τ) p/3,

√p

(13,17) 8.015 (13,19) 8.18141 6 4 (1,3) 8.062 13.667 , 6.403 (17,19) 8.311

5 (1,3) 7.797 43 3 3 (1,5) 8.544 14.333 , 6.557(1,7) 7.919 6 (1,5) 6.557(1,9) 8.402 7 (1,5) 8.426(3,7) 8.363 (1,11) 7.963(3,9) 7.919 (1,13) 8.484(7,9) 7.797 (1,17) 8.426

8 (1,3) 8.394 (1,23) 8.302(1,7) 6.403 (5,11) 8.426(1,13) 8.224 (5,13) 8.302(3,7) 8.224 (5,17) 8.26(3,13) 6.403 (5,23) 8.537(7,13) 8.394 (11,13) 8.426

10 (1,3) 8.139 (11,17) 8.502(1,7) 7.919 (11,23) 8.26(1,9) 8.363 (13,17) 7.963(3,7) 8.402 (13,23) 8.426(3,9) 7.919 (17,23) 8.426(7,9) 8.139 14 (1,5) 8.295

20 (1,3) 8.311 (1,11) 8.408(1,7) 8.181 (1,13) 6.557(1,9) 8.363 (1,17) 8.386(1,11) 8.307 (1,23) 8.505(1,13) 8.212 (5,11) 8.027(1,17) 7.974 (5,13) 8.505(1,19) 8.402 (5,17) 8.386(3,7) 8.402 (5,23) 6.557(3,9) 8.181 (11,13) 8.386(3,11) 8.347 (11,17) 6.557(3,13) 8.307 (11,23) 8.386(3,17) 8.402 (13,17) 8.408(3,19) 7.974 (13,23) 8.295(7,9) 8.311 (17,23) 8.027(7,11) 8.251 21 (1,5) 8.35(7,13) 8.402 (1,11) 8.535(7,17) 8.307 (1,13) 8.426(7,19) 8.212 (1,17) 8.35(9,11) 8.402 (1,19) 8.143(9,13) 8.251 (1,23) 8.361(9,17) 8.347 (1,25) 8.46(9,19) 8.307 (1,29) 8.355(11,13) 8.311 (1,31) 7.78(11,17) 8.181 (1,37) 8.305(11,19) 8.363 (1,41) 8.545(13,17) 8.402 (5,11) 8.182


39


τC(τ) p/3,


τC(τ) p/3,

√p

(5,13) 8.334 (25,37) 8.437(5,17) 8.437 (25,41) 8.35(5,19) 8.275 (29,31) 8.35(5,23) 8.426 (29,37) 8.334(5,25) 8.35 (29,41) 8.426(5,29) 8.002 (31,37) 8.182(5,31) 8.554 (31,41) 8.535(5,37) 8.555 (37,41) 8.35(5,41) 8.305 47 5 23 (1,3) 8.64 15.667 , 6.856(11,13) 8.35 (1,5) 8.756(11,17) 8.426 (1,7) 8.835(11,19) 8.35 (1,9) 8.653(11,23) 8.421 (1,11) 8.376(11,25) 8.246 (1,13) 8.65(11,29) 8.434 (1,15) 8.688(11,31) 8.554 (1,17) 8.825(11,37) 8.554 (1,19) 8.756(11,41) 7.78 (1,21) 8.78(13,17) 8.53 (1,25) 8.6(13,19) 8.508 (1,27) 8.777(13,23) 8.35 (1,29) 8.8(13,25) 7.955 (1,31) 8.514(13,29) 8.548 (1,33) 8.653(13,31) 8.434 (1,35) 8.791(13,37) 8.002 (1,37) 8.825(13,41) 8.355 (1,39) 8.734(17,19) 8.506 (1,41) 8.828(17,23) 8.555 (1,43) 8.828(17,25) 8.556 (1,45) 8.847(17,29) 7.955 (3,5) 8.769(17,31) 8.246 (3,7) 8.663(17,37) 8.35 (3,9) 8.581(17,41) 8.46 (3,11) 8.832(19,23) 8.551 (3,13) 8.636(19,25) 8.555 (3,15) 8.825(19,29) 8.35 (3,17) 8.6(19,31) 8.421 (3,19) 8.772(19,37) 8.426 (3,21) 8.82(19,41) 8.361 (3,25) 8.741(23,25) 8.506 (3,27) 8.777(23,29) 8.508 (3,29) 8.816(23,31) 8.35 (3,31) 8.755(23,37) 8.275 (3,33) 8.456(23,41) 8.143 (3,35) 8.616(25,29) 8.53 (3,37) 8.756(25,31) 8.426 (3,39) 8.765


40


τC(τ) p/3,


τC(τ) p/3,

√p

(3,41) 8.828 (9,25) 8.727(3,43) 8.847 (9,27) 8.516(3,45) 8.828 (9,29) 8.569(5,7) 8.64 (9,31) 8.755(5,9) 8.541 (9,33) 8.822(5,11) 8.717 (9,35) 8.664(5,13) 8.837 (9,37) 8.852(5,15) 8.784 (9,39) 8.805(5,17) 8.758 (9,41) 8.848(5,19) 8.62 (9,43) 8.756(5,21) 8.746 (9,45) 8.825(5,25) 8.68 (11,13) 8.69(5,27) 8.69 (11,15) 8.755(5,29) 8.781 (11,17) 8.744(5,31) 8.741 (11,19) 8.78(5,33) 8.793 (11,21) 8.67(5,35) 8.52 (11,25) 8.564(5,37) 8.848 (11,27) 8.639(5,39) 8.688 (11,29) 8.83(5,41) 8.848 (11,31) 8.763(5,43) 8.828 (11,33) 8.717(5,45) 8.828 (11,35) 8.853(7,9) 8.774 (11,37) 8.664(7,11) 8.742 (11,39) 8.699(7,13) 8.774 (11,41) 8.52(7,15) 8.548 (11,43) 8.616(7,17) 8.64 (11,45) 8.791(7,19) 8.631 (13,15) 8.669(7,21) 8.527 (13,17) 8.739(7,25) 8.663 (13,19) 8.775(7,27) 8.769 (13,21) 8.571(7,29) 8.662 (13,25) 8.657(7,31) 8.441 (13,27) 8.62(7,33) 8.629 (13,29) 8.777(7,35) 8.699 (13,31) 8.667(7,37) 8.805 (13,33) 8.847(7,39) 8.856 (13,35) 8.717(7,41) 8.688 (13,37) 8.822(7,43) 8.765 (13,39) 8.629(7,45) 8.734 (13,41) 8.793(9,11) 8.749 (13,43) 8.456(9,13) 8.785 (13,45) 8.653(9,15) 8.749 (15,17) 8.687(9,17) 8.825 (15,19) 8.441(9,19) 8.69 (15,21) 8.832(9,21) 8.568 (15,25) 8.674


41


τC(τ) p/3,


τC(τ) p/3,

√p

(15,27) 8.554 (25,27) 8.347(15,29) 8.781 (25,29) 8.832(15,31) 8.854 (25,31) 8.832(15,33) 8.667 (25,33) 8.571(15,35) 8.763 (25,35) 8.67(15,37) 8.755 (25,37) 8.568(15,39) 8.441 (25,39) 8.527(15,41) 8.741 (25,41) 8.746(15,43) 8.755 (25,43) 8.82(15,45) 8.514 (25,45) 8.78(17,19) 8.78 (27,29) 8.78(17,21) 8.832 (27,31) 8.441(17,25) 8.56 (27,33) 8.775(17,27) 8.765 (27,35) 8.78(17,29) 8.855 (27,37) 8.69(17,31) 8.781 (27,39) 8.631(17,33) 8.777 (27,41) 8.62(17,35) 8.83 (27,43) 8.772(17,37) 8.569 (27,45) 8.756(17,39) 8.662 (29,31) 8.687(17,41) 8.781 (29,33) 8.739(17,43) 8.816 (29,35) 8.744(17,45) 8.8 (29,37) 8.825(19,21) 8.347 (29,39) 8.64(19,25) 8.781 (29,41) 8.758(19,27) 8.854 (29,43) 8.6(19,29) 8.765 (29,45) 8.825(19,31) 8.554 (31,33) 8.669(19,33) 8.62 (31,35) 8.755(19,35) 8.639 (31,37) 8.749(19,37) 8.516 (31,39) 8.548(19,39) 8.769 (31,41) 8.784(19,41) 8.69 (31,43) 8.825(19,43) 8.777 (31,45) 8.688(19,45) 8.777 (33,35) 8.69(21,25) 8.844 (33,37) 8.785(21,27) 8.781 (33,39) 8.774(21,29) 8.56 (33,41) 8.837(21,31) 8.674 (33,43) 8.636(21,33) 8.657 (33,45) 8.65(21,35) 8.564 (35,37) 8.749(21,37) 8.727 (35,39) 8.742(21,39) 8.663 (35,41) 8.717(21,41) 8.68 (35,43) 8.832(21,43) 8.741 (35,45) 8.376(21,45) 8.6 (37,39) 8.774


42


τC(τ) p/3,


τC(τ) p/3,

√p

(37,41) 8.541 (7,23) 9.28(37,43) 8.581 (7,25) 8.776(37,45) 8.653 (9,11) 8.834(39,41) 8.64 (9,15) 8.766(39,43) 8.663 (9,17) 9.28(39,45) 8.835 (9,19) 9.009(41,43) 8.769 (9,21) 9.22(41,45) 8.756 (9,23) 9.036(43,45) 8.64 (9,25) 9.036

53 2 4 (1,3) 7.28 17.667 , 7.28 (11,15) 9.27813 (1,3) 9.045 (11,17) 8.766

(1,5) 8.892 (11,19) 8.775(1,7) 8.774 (11,21) 9.174(1,9) 9.079 (11,23) 9.129(1,11) 8.881 (11,25) 8.854(1,15) 8.854 (15,17) 8.834(1,17) 9.036 (15,19) 9.129(1,19) 8.776 (15,21) 8.775(1,21) 8.945 (15,23) 9.043(1,23) 9.036 (15,25) 8.881(1,25) 9.263 (17,19) 8.991(3,5) 8.536 (17,21) 9.171(3,7) 9.16 (17,23) 8.932(3,9) 8.932 (17,25) 9.079(3,11) 9.043 (19,21) 8.775(3,15) 9.129 (19,23) 9.16(3,17) 9.036 (19,25) 8.774(3,19) 9.28 (21,23) 8.536(3,21) 8.67 (21,25) 8.892(3,23) 9.24 (23,25) 9.045(3,25) 9.036 26 (1,3) 9.139(5,7) 8.775 (1,5) 9.129(5,9) 9.171 (1,7) 9.108(5,11) 8.775 (1,9) 9.278(5,15) 9.174 (1,11) 9.044(5,17) 9.22 (1,15) 9.01(5,19) 8.927 (1,17) 9.034(5,21) 9.28 (1,19) 9.254(5,23) 8.67 (1,21) 9.214(5,25) 8.945 (1,23) 8.81(7,9) 8.991 (1,25) 9.278(7,11) 9.129 (3,5) 9.085(7,15) 8.775 (3,7) 9.134(7,17) 9.009 (3,9) 9.151(7,19) 9.28 (3,11) 9.073(7,21) 8.927 (3,15) 9.212


43


τC(τ) p/3,


τC(τ) p/3,

√p

(3,17) 9.221 (21,23) 9.085(3,19) 9.222 (21,25) 9.129(3,21) 9.28 (23,25) 9.139(3,23) 9.28(3,25) 8.81(5,7) 9.02(5,9) 9.28(5,11) 8.807(5,15) 8.979(5,17) 9.264(5,19) 9.219(5,21) 9.28(5,23) 9.28(5,25) 9.214(7,11) 8.969(7,15) 9.22(7,17) 9.246(7,19) 9.273(7,21) 9.219(7,23) 9.222(7,25) 9.254(9,11) 9.18(9,15) 9.153(9,17) 9.28(9,19) 9.246(9,21) 9.264(9,23) 9.221(9,25) 9.034(11,15) 9.28(11,17) 9.153(11,19) 9.22(11,21) 8.979(11,23) 9.212(11,25) 9.01(15,17) 9.18(15,19) 8.969(15,21) 8.807(15,23) 9.073(15,25) 9.044(17,19) 9.196(17,21) 9.28(17,23) 9.151(17,25) 9.278(19,21) 9.02(19,23) 9.134(19,25) 9.108

44

Table 5.2: Two-level Crosscorrelation pairs in A(M(s(n))) p ≤ 97.

p(√

p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

13 4 (1 0 0),(3 0 0) 3.606 (1 0 4),(5 0 5) 4.359(1 0 0),(3 0 1) 3.606 (1 0 5),(5 0 0) 4.359

(3.606) (1 0 0),(3 0 2) 3.606 (1 0 5),(5 0 1) 4.359(1 0 0),(3 0 3) 3.606 (1 0 5),(5 0 2) 4.359(1 0 1),(3 0 0) 3.606 (1 0 5),(5 0 3) 4.359(1 0 1),(3 0 1) 3.606 (1 0 5),(5 0 4) 4.359(1 0 1),(3 0 2) 3.606 (1 0 5),(5 0 5) 4.359(1 0 1),(3 0 3) 3.606 29 4 (1 0 0),(3 0 0) 5.385(1 0 2),(3 0 0) 3.606 (1 0 0),(3 0 1) 5.385(1 0 2),(3 0 1) 3.606 (5.385) (1 0 0),(3 0 2) 5.385(1 0 2),(3 0 2) 3.606 (1 0 0),(3 0 3) 5.385(1 0 2),(3 0 3) 3.606 (1 0 1),(3 0 0) 5.385(1 0 3),(3 0 0) 3.606 (1 0 1),(3 0 1) 5.385(1 0 3),(3 0 1) 3.606 (1 0 1),(3 0 2) 5.385(1 0 3),(3 0 2) 3.606 (1 0 1),(3 0 3) 5.385(1 0 3),(3 0 3) 3.606 (1 0 2),(3 0 0) 5.385

19 6 (1 0 0),(5 0 0) 4.359 (1 0 2),(3 0 1) 5.385(1 0 0),(5 0 1) 4.359 (1 0 2),(3 0 2) 5.385

(4.359) (1 0 0),(5 0 2) 4.359 (1 0 2),(3 0 3) 5.385(1 0 0),(5 0 3) 4.359 (1 0 3),(3 0 0) 5.385(1 0 0),(5 0 4) 4.359 (1 0 3),(3 0 1) 5.385(1 0 0),(5 0 5) 4.359 (1 0 3),(3 0 2) 5.385(1 0 1),(5 0 0) 4.359 (1 0 3),(3 0 3) 5.385(1 0 1),(5 0 1) 4.359 31 6 (1 0 0),(5 0 0) 5.568(1 0 1),(5 0 2) 4.359 (1 0 0),(5 0 1) 5.568(1 0 1),(5 0 3) 4.359 (5.568) (1 0 0),(5 0 2) 5.568(1 0 1),(5 0 4) 4.359 (1 0 0),(5 0 3) 5.568(1 0 1),(5 0 5) 4.359 (1 0 0),(5 0 4) 5.568(1 0 2),(5 0 0) 4.359 (1 0 0),(5 0 5) 5.568(1 0 2),(5 0 1) 4.359 (1 0 1),(5 0 0) 5.568(1 0 2),(5 0 2) 4.359 (1 0 1),(5 0 1) 5.568(1 0 2),(5 0 3) 4.359 (1 0 1),(5 0 2) 5.568(1 0 2),(5 0 4) 4.359 (1 0 1),(5 0 3) 5.568(1 0 2),(5 0 5) 4.359 (1 0 1),(5 0 4) 5.568(1 0 3),(5 0 0) 4.359 (1 0 1),(5 0 5) 5.568(1 0 3),(5 0 1) 4.359 (1 0 2),(5 0 0) 5.568(1 0 3),(5 0 2) 4.359 (1 0 2),(5 0 1) 5.568(1 0 3),(5 0 3) 4.359 (1 0 2),(5 0 2) 5.568(1 0 3),(5 0 4) 4.359 (1 0 2),(5 0 3) 5.568(1 0 3),(5 0 5) 4.359 (1 0 2),(5 0 4) 5.568(1 0 4),(5 0 0) 4.359 (1 0 2),(5 0 5) 5.568(1 0 4),(5 0 1) 4.359 (1 0 3),(5 0 0) 5.568(1 0 4),(5 0 2) 4.359 (1 0 3),(5 0 1) 5.568(1 0 4),(5 0 3) 4.359 (1 0 3),(5 0 2) 5.568(1 0 4),(5 0 4) 4.359 (1 0 3),(5 0 3) 5.568

continued on next page

45

continued from previous pagep(√

p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 3),(5 0 4) 5.568 (1 0 3),(9 0 2) 5.568(1 0 3),(5 0 5) 5.568 (1 0 3),(9 0 3) 5.568(1 0 4),(5 0 0) 5.568 (1 0 3),(9 0 4) 5.568(1 0 4),(5 0 1) 5.568 (1 0 3),(9 0 5) 5.568(1 0 4),(5 0 2) 5.568 (1 0 3),(9 0 6) 5.568(1 0 4),(5 0 3) 5.568 (1 0 3),(9 0 7) 5.568(1 0 4),(5 0 4) 5.568 (1 0 3),(9 0 8) 5.568(1 0 4),(5 0 5) 5.568 (1 0 3),(9 0 9) 5.568(1 0 5),(5 0 0) 5.568 (1 0 4),(9 0 0) 5.568(1 0 5),(5 0 1) 5.568 (1 0 4),(9 0 1) 5.568(1 0 5),(5 0 2) 5.568 (1 0 4),(9 0 2) 5.568(1 0 5),(5 0 3) 5.568 (1 0 4),(9 0 3) 5.568(1 0 5),(5 0 4) 5.568 (1 0 4),(9 0 4) 5.568(1 0 5),(5 0 5) 5.568 (1 0 4),(9 0 5) 5.568

10 (1 0 0),(9 0 0) 5.568 (1 0 4),(9 0 6) 5.568(1 0 0),(9 0 1) 5.568 (1 0 4),(9 0 7) 5.568(1 0 0),(9 0 2) 5.568 (1 0 4),(9 0 8) 5.568(1 0 0),(9 0 3) 5.568 (1 0 4),(9 0 9) 5.568(1 0 0),(9 0 4) 5.568 (1 0 5),(9 0 0) 5.568(1 0 0),(9 0 5) 5.568 (1 0 5),(9 0 1) 5.568(1 0 0),(9 0 6) 5.568 (1 0 5),(9 0 2) 5.568(1 0 0),(9 0 7) 5.568 (1 0 5),(9 0 3) 5.568(1 0 0),(9 0 8) 5.568 (1 0 5),(9 0 4) 5.568(1 0 0),(9 0 9) 5.568 (1 0 5),(9 0 5) 5.568(1 0 1),(9 0 0) 5.568 (1 0 5),(9 0 6) 5.568(1 0 1),(9 0 1) 5.568 (1 0 5),(9 0 7) 5.568(1 0 1),(9 0 2) 5.568 (1 0 5),(9 0 8) 5.568(1 0 1),(9 0 3) 5.568 (1 0 5),(9 0 9) 5.568(1 0 1),(9 0 4) 5.568 (1 0 6),(9 0 0) 5.568(1 0 1),(9 0 5) 5.568 (1 0 6),(9 0 1) 5.568(1 0 1),(9 0 6) 5.568 (1 0 6),(9 0 2) 5.568(1 0 1),(9 0 7) 5.568 (1 0 6),(9 0 3) 5.568(1 0 1),(9 0 8) 5.568 (1 0 6),(9 0 4) 5.568(1 0 1),(9 0 9) 5.568 (1 0 6),(9 0 5) 5.568(1 0 2),(9 0 0) 5.568 (1 0 6),(9 0 6) 5.568(1 0 2),(9 0 1) 5.568 (1 0 6),(9 0 7) 5.568(1 0 2),(9 0 2) 5.568 (1 0 6),(9 0 8) 5.568(1 0 2),(9 0 3) 5.568 (1 0 6),(9 0 9) 5.568(1 0 2),(9 0 4) 5.568 (1 0 7),(9 0 0) 5.568(1 0 2),(9 0 5) 5.568 (1 0 7),(9 0 1) 5.568(1 0 2),(9 0 6) 5.568 (1 0 7),(9 0 2) 5.568(1 0 2),(9 0 7) 5.568 (1 0 7),(9 0 3) 5.568(1 0 2),(9 0 8) 5.568 (1 0 7),(9 0 4) 5.568(1 0 2),(9 0 9) 5.568 (1 0 7),(9 0 5) 5.568(1 0 3),(9 0 0) 5.568 (1 0 7),(9 0 6) 5.568(1 0 3),(9 0 1) 5.568 (1 0 7),(9 0 7) 5.568


46


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 7),(9 0 8) 5.568 (3 0 2),(7 0 4) 5.568(1 0 7),(9 0 9) 5.568 (3 0 2),(7 0 5) 5.568(1 0 8),(9 0 0) 5.568 (3 0 2),(7 0 6) 5.568(1 0 8),(9 0 1) 5.568 (3 0 2),(7 0 7) 5.568(1 0 8),(9 0 2) 5.568 (3 0 2),(7 0 8) 5.568(1 0 8),(9 0 3) 5.568 (3 0 2),(7 0 9) 5.568(1 0 8),(9 0 4) 5.568 (3 0 3),(7 0 0) 5.568(1 0 8),(9 0 5) 5.568 (3 0 3),(7 0 1) 5.568(1 0 8),(9 0 6) 5.568 (3 0 3),(7 0 2) 5.568(1 0 8),(9 0 7) 5.568 (3 0 3),(7 0 3) 5.568(1 0 8),(9 0 8) 5.568 (3 0 3),(7 0 4) 5.568(1 0 8),(9 0 9) 5.568 (3 0 3),(7 0 5) 5.568(1 0 9),(9 0 0) 5.568 (3 0 3),(7 0 6) 5.568(1 0 9),(9 0 1) 5.568 (3 0 3),(7 0 7) 5.568(1 0 9),(9 0 2) 5.568 (3 0 3),(7 0 8) 5.568(1 0 9),(9 0 3) 5.568 (3 0 3),(7 0 9) 5.568(1 0 9),(9 0 4) 5.568 (3 0 4),(7 0 0) 5.568(1 0 9),(9 0 5) 5.568 (3 0 4),(7 0 1) 5.568(1 0 9),(9 0 6) 5.568 (3 0 4),(7 0 2) 5.568(1 0 9),(9 0 7) 5.568 (3 0 4),(7 0 3) 5.568(1 0 9),(9 0 8) 5.568 (3 0 4),(7 0 4) 5.568(1 0 9),(9 0 9) 5.568 (3 0 4),(7 0 5) 5.568(3 0 0),(7 0 0) 5.568 (3 0 4),(7 0 6) 5.568(3 0 0),(7 0 1) 5.568 (3 0 4),(7 0 7) 5.568(3 0 0),(7 0 2) 5.568 (3 0 4),(7 0 8) 5.568(3 0 0),(7 0 3) 5.568 (3 0 4),(7 0 9) 5.568(3 0 0),(7 0 4) 5.568 (3 0 5),(7 0 0) 5.568(3 0 0),(7 0 5) 5.568 (3 0 5),(7 0 1) 5.568(3 0 0),(7 0 6) 5.568 (3 0 5),(7 0 2) 5.568(3 0 0),(7 0 7) 5.568 (3 0 5),(7 0 3) 5.568(3 0 0),(7 0 8) 5.568 (3 0 5),(7 0 4) 5.568(3 0 0),(7 0 9) 5.568 (3 0 5),(7 0 5) 5.568(3 0 1),(7 0 0) 5.568 (3 0 5),(7 0 6) 5.568(3 0 1),(7 0 1) 5.568 (3 0 5),(7 0 7) 5.568(3 0 1),(7 0 2) 5.568 (3 0 5),(7 0 8) 5.568(3 0 1),(7 0 3) 5.568 (3 0 5),(7 0 9) 5.568(3 0 1),(7 0 4) 5.568 (3 0 6),(7 0 0) 5.568(3 0 1),(7 0 5) 5.568 (3 0 6),(7 0 1) 5.568(3 0 1),(7 0 6) 5.568 (3 0 6),(7 0 2) 5.568(3 0 1),(7 0 7) 5.568 (3 0 6),(7 0 3) 5.568(3 0 1),(7 0 8) 5.568 (3 0 6),(7 0 4) 5.568(3 0 1),(7 0 9) 5.568 (3 0 6),(7 0 5) 5.568(3 0 2),(7 0 0) 5.568 (3 0 6),(7 0 6) 5.568(3 0 2),(7 0 1) 5.568 (3 0 6),(7 0 7) 5.568(3 0 2),(7 0 2) 5.568 (3 0 6),(7 0 8) 5.568(3 0 2),(7 0 3) 5.568 (3 0 6),(7 0 9) 5.568


47


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(3 0 7),(7 0 0) 5.568 41 8 (1 0 0),(7 0 0) 6.403(3 0 7),(7 0 1) 5.568 (1 0 0),(7 0 1) 6.403(3 0 7),(7 0 2) 5.568 (6.403) (1 0 0),(7 0 2) 6.403(3 0 7),(7 0 3) 5.568 (1 0 0),(7 0 3) 6.403(3 0 7),(7 0 4) 5.568 (1 0 0),(7 0 4) 6.403(3 0 7),(7 0 5) 5.568 (1 0 0),(7 0 5) 6.403(3 0 7),(7 0 6) 5.568 (1 0 0),(7 0 6) 6.403(3 0 7),(7 0 7) 5.568 (1 0 0),(7 0 7) 6.403(3 0 7),(7 0 8) 5.568 (1 0 1),(7 0 0) 6.403(3 0 7),(7 0 9) 5.568 (1 0 1),(7 0 1) 6.403(3 0 8),(7 0 0) 5.568 (1 0 1),(7 0 2) 6.403(3 0 8),(7 0 1) 5.568 (1 0 1),(7 0 3) 6.403(3 0 8),(7 0 2) 5.568 (1 0 1),(7 0 4) 6.403(3 0 8),(7 0 3) 5.568 (1 0 1),(7 0 5) 6.403(3 0 8),(7 0 4) 5.568 (1 0 1),(7 0 6) 6.403(3 0 8),(7 0 5) 5.568 (1 0 1),(7 0 7) 6.403(3 0 8),(7 0 6) 5.568 (1 0 2),(7 0 0) 6.403(3 0 8),(7 0 7) 5.568 (1 0 2),(7 0 1) 6.403(3 0 8),(7 0 8) 5.568 (1 0 2),(7 0 2) 6.403(3 0 8),(7 0 9) 5.568 (1 0 2),(7 0 3) 6.403(3 0 9),(7 0 0) 5.568 (1 0 2),(7 0 4) 6.403(3 0 9),(7 0 1) 5.568 (1 0 2),(7 0 5) 6.403(3 0 9),(7 0 2) 5.568 (1 0 2),(7 0 6) 6.403(3 0 9),(7 0 3) 5.568 (1 0 2),(7 0 7) 6.403(3 0 9),(7 0 4) 5.568 (1 0 3),(7 0 0) 6.403(3 0 9),(7 0 5) 5.568 (1 0 3),(7 0 1) 6.403(3 0 9),(7 0 6) 5.568 (1 0 3),(7 0 2) 6.403(3 0 9),(7 0 7) 5.568 (1 0 3),(7 0 3) 6.403(3 0 9),(7 0 8) 5.568 (1 0 3),(7 0 4) 6.403(3 0 9),(7 0 9) 5.568 (1 0 3),(7 0 5) 6.403

37 4 (1 0 0),(3 0 0) 6.083 (1 0 3),(7 0 6) 6.403(1 0 0),(3 0 1) 6.083 (1 0 3),(7 0 7) 6.403

(6.083) (1 0 0),(3 0 2) 6.083 (1 0 4),(7 0 0) 6.403(1 0 0),(3 0 3) 6.083 (1 0 4),(7 0 1) 6.403(1 0 1),(3 0 0) 6.083 (1 0 4),(7 0 2) 6.403(1 0 1),(3 0 1) 6.083 (1 0 4),(7 0 3) 6.403(1 0 1),(3 0 2) 6.083 (1 0 4),(7 0 4) 6.403(1 0 1),(3 0 3) 6.083 (1 0 4),(7 0 5) 6.403(1 0 2),(3 0 0) 6.083 (1 0 4),(7 0 6) 6.403(1 0 2),(3 0 1) 6.083 (1 0 4),(7 0 7) 6.403(1 0 2),(3 0 2) 6.083 (1 0 5),(7 0 0) 6.403(1 0 2),(3 0 3) 6.083 (1 0 5),(7 0 1) 6.403(1 0 3),(3 0 0) 6.083 (1 0 5),(7 0 2) 6.403(1 0 3),(3 0 1) 6.083 (1 0 5),(7 0 3) 6.403(1 0 3),(3 0 2) 6.083 (1 0 5),(7 0 4) 6.403(1 0 3),(3 0 3) 6.083 (1 0 5),(7 0 5) 6.403


48


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 5),(7 0 6) 6.403 (3 0 3),(5 0 4) 6.403(1 0 5),(7 0 7) 6.403 (3 0 3),(5 0 5) 6.403(1 0 6),(7 0 0) 6.403 (3 0 3),(5 0 6) 6.403(1 0 6),(7 0 1) 6.403 (3 0 3),(5 0 7) 6.403(1 0 6),(7 0 2) 6.403 (3 0 4),(5 0 0) 6.403(1 0 6),(7 0 3) 6.403 (3 0 4),(5 0 1) 6.403(1 0 6),(7 0 4) 6.403 (3 0 4),(5 0 2) 6.403(1 0 6),(7 0 5) 6.403 (3 0 4),(5 0 3) 6.403(1 0 6),(7 0 6) 6.403 (3 0 4),(5 0 4) 6.403(1 0 6),(7 0 7) 6.403 (3 0 4),(5 0 5) 6.403(1 0 7),(7 0 0) 6.403 (3 0 4),(5 0 6) 6.403(1 0 7),(7 0 1) 6.403 (3 0 4),(5 0 7) 6.403(1 0 7),(7 0 2) 6.403 (3 0 5),(5 0 0) 6.403(1 0 7),(7 0 3) 6.403 (3 0 5),(5 0 1) 6.403(1 0 7),(7 0 4) 6.403 (3 0 5),(5 0 2) 6.403(1 0 7),(7 0 5) 6.403 (3 0 5),(5 0 3) 6.403(1 0 7),(7 0 6) 6.403 (3 0 5),(5 0 4) 6.403(1 0 7),(7 0 7) 6.403 (3 0 5),(5 0 5) 6.403(3 0 0),(5 0 0) 6.403 (3 0 5),(5 0 6) 6.403(3 0 0),(5 0 1) 6.403 (3 0 5),(5 0 7) 6.403(3 0 0),(5 0 2) 6.403 (3 0 6),(5 0 0) 6.403(3 0 0),(5 0 3) 6.403 (3 0 6),(5 0 1) 6.403(3 0 0),(5 0 4) 6.403 (3 0 6),(5 0 2) 6.403(3 0 0),(5 0 5) 6.403 (3 0 6),(5 0 3) 6.403(3 0 0),(5 0 6) 6.403 (3 0 6),(5 0 4) 6.403(3 0 0),(5 0 7) 6.403 (3 0 6),(5 0 5) 6.403(3 0 1),(5 0 0) 6.403 (3 0 6),(5 0 6) 6.403(3 0 1),(5 0 1) 6.403 (3 0 6),(5 0 7) 6.403(3 0 1),(5 0 2) 6.403 (3 0 7),(5 0 0) 6.403(3 0 1),(5 0 3) 6.403 (3 0 7),(5 0 1) 6.403(3 0 1),(5 0 4) 6.403 (3 0 7),(5 0 2) 6.403(3 0 1),(5 0 5) 6.403 (3 0 7),(5 0 3) 6.403(3 0 1),(5 0 6) 6.403 (3 0 7),(5 0 4) 6.403(3 0 1),(5 0 7) 6.403 (3 0 7),(5 0 5) 6.403(3 0 2),(5 0 0) 6.403 (3 0 7),(5 0 6) 6.403(3 0 2),(5 0 1) 6.403 (3 0 7),(5 0 7) 6.403(3 0 2),(5 0 2) 6.403 43 6 (1 0 0),(5 0 0) 6.557(3 0 2),(5 0 3) 6.403 (1 0 0),(5 0 1) 6.557(3 0 2),(5 0 4) 6.403 (6.557) (1 0 0),(5 0 2) 6.557(3 0 2),(5 0 5) 6.403 (1 0 0),(5 0 3) 6.557(3 0 2),(5 0 6) 6.403 (1 0 0),(5 0 4) 6.557(3 0 2),(5 0 7) 6.403 (1 0 0),(5 0 5) 6.557(3 0 3),(5 0 0) 6.403 (1 0 1),(5 0 0) 6.557(3 0 3),(5 0 1) 6.403 (1 0 1),(5 0 1) 6.557(3 0 3),(5 0 2) 6.403 (1 0 1),(5 0 2) 6.557(3 0 3),(5 0 3) 6.403 (1 0 1),(5 0 3) 6.557


49


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 1),(5 0 4) 6.557 (1 0 1),(3 0 0) 7.81(1 0 1),(5 0 5) 6.557 (1 0 1),(3 0 1) 7.81(1 0 2),(5 0 0) 6.557 (1 0 1),(3 0 2) 7.81(1 0 2),(5 0 1) 6.557 (1 0 1),(3 0 3) 7.81(1 0 2),(5 0 2) 6.557 (1 0 2),(3 0 0) 7.81(1 0 2),(5 0 3) 6.557 (1 0 2),(3 0 1) 7.81(1 0 2),(5 0 4) 6.557 (1 0 2),(3 0 2) 7.81(1 0 2),(5 0 5) 6.557 (1 0 2),(3 0 3) 7.81(1 0 3),(5 0 0) 6.557 (1 0 3),(3 0 0) 7.81(1 0 3),(5 0 1) 6.557 (1 0 3),(3 0 1) 7.81(1 0 3),(5 0 2) 6.557 (1 0 3),(3 0 2) 7.81(1 0 3),(5 0 3) 6.557 (1 0 3),(3 0 3) 7.81(1 0 3),(5 0 4) 6.557 67 6 (1 0 0),(5 0 0) 8.185(1 0 3),(5 0 5) 6.557 (1 0 0),(5 0 1) 8.185(1 0 4),(5 0 0) 6.557 (8.185) (1 0 0),(5 0 2) 8.185(1 0 4),(5 0 1) 6.557 (1 0 0),(5 0 3) 8.185(1 0 4),(5 0 2) 6.557 (1 0 0),(5 0 4) 8.185(1 0 4),(5 0 3) 6.557 (1 0 0),(5 0 5) 8.185(1 0 4),(5 0 4) 6.557 (1 0 1),(5 0 0) 8.185(1 0 4),(5 0 5) 6.557 (1 0 1),(5 0 1) 8.185(1 0 5),(5 0 0) 6.557 (1 0 1),(5 0 2) 8.185(1 0 5),(5 0 1) 6.557 (1 0 1),(5 0 3) 8.185(1 0 5),(5 0 2) 6.557 (1 0 1),(5 0 4) 8.185(1 0 5),(5 0 3) 6.557 (1 0 1),(5 0 5) 8.185(1 0 5),(5 0 4) 6.557 (1 0 2),(5 0 0) 8.185(1 0 5),(5 0 5) 6.557 (1 0 2),(5 0 1) 8.185

53 4 (1 0 0),(3 0 0) 7.28 (1 0 2),(5 0 2) 8.185(1 0 0),(3 0 1) 7.28 (1 0 2),(5 0 3) 8.185

(7.28) (1 0 0),(3 0 2) 7.28 (1 0 2),(5 0 4) 8.185(1 0 0),(3 0 3) 7.28 (1 0 2),(5 0 5) 8.185(1 0 1),(3 0 0) 7.28 (1 0 3),(5 0 0) 8.185(1 0 1),(3 0 1) 7.28 (1 0 3),(5 0 1) 8.185(1 0 1),(3 0 2) 7.28 (1 0 3),(5 0 2) 8.185(1 0 1),(3 0 3) 7.28 (1 0 3),(5 0 3) 8.185(1 0 2),(3 0 0) 7.28 (1 0 3),(5 0 4) 8.185(1 0 2),(3 0 1) 7.28 (1 0 3),(5 0 5) 8.185(1 0 2),(3 0 2) 7.28 (1 0 4),(5 0 0) 8.185(1 0 2),(3 0 3) 7.28 (1 0 4),(5 0 1) 8.185(1 0 3),(3 0 0) 7.28 (1 0 4),(5 0 2) 8.185(1 0 3),(3 0 1) 7.28 (1 0 4),(5 0 3) 8.185(1 0 3),(3 0 2) 7.28 (1 0 4),(5 0 4) 8.185(1 0 3),(3 0 3) 7.28 (1 0 4),(5 0 5) 8.185

61 4 (1 0 0),(3 0 0) 7.81 (1 0 5),(5 0 0) 8.185(1 0 0),(3 0 1) 7.81 (1 0 5),(5 0 1) 8.185

(7.81) (1 0 0),(3 0 2) 7.81 (1 0 5),(5 0 2) 8.185(1 0 0),(3 0 3) 7.81 (1 0 5),(5 0 3) 8.185


50


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 5),(5 0 4) 8.185 (1 0 4),(9 0 4) 8.426(1 0 5),(5 0 5) 8.185 (1 0 4),(9 0 5) 8.426

71 10 (1 0 0),(9 0 0) 8.426 (1 0 4),(9 0 6) 8.426(1 0 0),(9 0 1) 8.426 (1 0 4),(9 0 7) 8.426

(8.426) (1 0 0),(9 0 2) 8.426 (1 0 4),(9 0 8) 8.426(1 0 0),(9 0 3) 8.426 (1 0 4),(9 0 9) 8.426(1 0 0),(9 0 4) 8.426 (1 0 5),(9 0 0) 8.426(1 0 0),(9 0 5) 8.426 (1 0 5),(9 0 1) 8.426(1 0 0),(9 0 6) 8.426 (1 0 5),(9 0 2) 8.426(1 0 0),(9 0 7) 8.426 (1 0 5),(9 0 3) 8.426(1 0 0),(9 0 8) 8.426 (1 0 5),(9 0 4) 8.426(1 0 0),(9 0 9) 8.426 (1 0 5),(9 0 5) 8.426(1 0 1),(9 0 0) 8.426 (1 0 5),(9 0 6) 8.426(1 0 1),(9 0 1) 8.426 (1 0 5),(9 0 7) 8.426(1 0 1),(9 0 2) 8.426 (1 0 5),(9 0 8) 8.426(1 0 1),(9 0 3) 8.426 (1 0 5),(9 0 9) 8.426(1 0 1),(9 0 4) 8.426 (1 0 6),(9 0 0) 8.426(1 0 1),(9 0 5) 8.426 (1 0 6),(9 0 1) 8.426(1 0 1),(9 0 6) 8.426 (1 0 6),(9 0 2) 8.426(1 0 1),(9 0 7) 8.426 (1 0 6),(9 0 3) 8.426(1 0 1),(9 0 8) 8.426 (1 0 6),(9 0 4) 8.426(1 0 1),(9 0 9) 8.426 (1 0 6),(9 0 5) 8.426(1 0 2),(9 0 0) 8.426 (1 0 6),(9 0 6) 8.426(1 0 2),(9 0 1) 8.426 (1 0 6),(9 0 7) 8.426(1 0 2),(9 0 2) 8.426 (1 0 6),(9 0 8) 8.426(1 0 2),(9 0 3) 8.426 (1 0 6),(9 0 9) 8.426(1 0 2),(9 0 4) 8.426 (1 0 7),(9 0 0) 8.426(1 0 2),(9 0 5) 8.426 (1 0 7),(9 0 1) 8.426(1 0 2),(9 0 6) 8.426 (1 0 7),(9 0 2) 8.426(1 0 2),(9 0 7) 8.426 (1 0 7),(9 0 3) 8.426(1 0 2),(9 0 8) 8.426 (1 0 7),(9 0 4) 8.426(1 0 2),(9 0 9) 8.426 (1 0 7),(9 0 5) 8.426(1 0 3),(9 0 0) 8.426 (1 0 7),(9 0 6) 8.426(1 0 3),(9 0 1) 8.426 (1 0 7),(9 0 7) 8.426(1 0 3),(9 0 2) 8.426 (1 0 7),(9 0 8) 8.426(1 0 3),(9 0 3) 8.426 (1 0 7),(9 0 9) 8.426(1 0 3),(9 0 4) 8.426 (1 0 8),(9 0 0) 8.426(1 0 3),(9 0 5) 8.426 (1 0 8),(9 0 1) 8.426(1 0 3),(9 0 6) 8.426 (1 0 8),(9 0 2) 8.426(1 0 3),(9 0 7) 8.426 (1 0 8),(9 0 3) 8.426(1 0 3),(9 0 8) 8.426 (1 0 8),(9 0 4) 8.426(1 0 3),(9 0 9) 8.426 (1 0 8),(9 0 5) 8.426(1 0 4),(9 0 0) 8.426 (1 0 8),(9 0 6) 8.426(1 0 4),(9 0 1) 8.426 (1 0 8),(9 0 7) 8.426(1 0 4),(9 0 2) 8.426 (1 0 8),(9 0 8) 8.426(1 0 4),(9 0 3) 8.426 (1 0 8),(9 0 9) 8.426


51


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 9),(9 0 0) 8.426 (3 0 3),(7 0 6) 8.426(1 0 9),(9 0 1) 8.426 (3 0 3),(7 0 7) 8.426(1 0 9),(9 0 2) 8.426 (3 0 3),(7 0 8) 8.426(1 0 9),(9 0 3) 8.426 (3 0 3),(7 0 9) 8.426(1 0 9),(9 0 4) 8.426 (3 0 4),(7 0 0) 8.426(1 0 9),(9 0 5) 8.426 (3 0 4),(7 0 1) 8.426(1 0 9),(9 0 6) 8.426 (3 0 4),(7 0 2) 8.426(1 0 9),(9 0 7) 8.426 (3 0 4),(7 0 3) 8.426(1 0 9),(9 0 8) 8.426 (3 0 4),(7 0 4) 8.426(1 0 9),(9 0 9) 8.426 (3 0 4),(7 0 5) 8.426(3 0 0),(7 0 0) 8.426 (3 0 4),(7 0 6) 8.426(3 0 0),(7 0 1) 8.426 (3 0 4),(7 0 7) 8.426(3 0 0),(7 0 2) 8.426 (3 0 4),(7 0 8) 8.426(3 0 0),(7 0 3) 8.426 (3 0 4),(7 0 9) 8.426(3 0 0),(7 0 4) 8.426 (3 0 5),(7 0 0) 8.426(3 0 0),(7 0 5) 8.426 (3 0 5),(7 0 1) 8.426(3 0 0),(7 0 6) 8.426 (3 0 5),(7 0 2) 8.426(3 0 0),(7 0 7) 8.426 (3 0 5),(7 0 3) 8.426(3 0 0),(7 0 8) 8.426 (3 0 5),(7 0 4) 8.426(3 0 0),(7 0 9) 8.426 (3 0 5),(7 0 5) 8.426(3 0 1),(7 0 0) 8.426 (3 0 5),(7 0 6) 8.426(3 0 1),(7 0 1) 8.426 (3 0 5),(7 0 7) 8.426(3 0 1),(7 0 2) 8.426 (3 0 5),(7 0 8) 8.426(3 0 1),(7 0 3) 8.426 (3 0 5),(7 0 9) 8.426(3 0 1),(7 0 4) 8.426 (3 0 6),(7 0 0) 8.426(3 0 1),(7 0 5) 8.426 (3 0 6),(7 0 1) 8.426(3 0 1),(7 0 6) 8.426 (3 0 6),(7 0 2) 8.426(3 0 1),(7 0 7) 8.426 (3 0 6),(7 0 3) 8.426(3 0 1),(7 0 8) 8.426 (3 0 6),(7 0 4) 8.426(3 0 1),(7 0 9) 8.426 (3 0 6),(7 0 5) 8.426(3 0 2),(7 0 0) 8.426 (3 0 6),(7 0 6) 8.426(3 0 2),(7 0 1) 8.426 (3 0 6),(7 0 7) 8.426(3 0 2),(7 0 2) 8.426 (3 0 6),(7 0 8) 8.426(3 0 2),(7 0 3) 8.426 (3 0 6),(7 0 9) 8.426(3 0 2),(7 0 4) 8.426 (3 0 7),(7 0 0) 8.426(3 0 2),(7 0 5) 8.426 (3 0 7),(7 0 1) 8.426(3 0 2),(7 0 6) 8.426 (3 0 7),(7 0 2) 8.426(3 0 2),(7 0 7) 8.426 (3 0 7),(7 0 3) 8.426(3 0 2),(7 0 8) 8.426 (3 0 7),(7 0 4) 8.426(3 0 2),(7 0 9) 8.426 (3 0 7),(7 0 5) 8.426(3 0 3),(7 0 0) 8.426 (3 0 7),(7 0 6) 8.426(3 0 3),(7 0 1) 8.426 (3 0 7),(7 0 7) 8.426(3 0 3),(7 0 2) 8.426 (3 0 7),(7 0 8) 8.426(3 0 3),(7 0 3) 8.426 (3 0 7),(7 0 9) 8.426(3 0 3),(7 0 4) 8.426 (3 0 8),(7 0 0) 8.426(3 0 3),(7 0 5) 8.426 (3 0 8),(7 0 1) 8.426


52


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(3 0 8),(7 0 2) 8.426 (1 0 3),(7 0 4) 8.544(3 0 8),(7 0 3) 8.426 (1 0 3),(7 0 5) 8.544(3 0 8),(7 0 4) 8.426 (1 0 3),(7 0 6) 8.544(3 0 8),(7 0 5) 8.426 (1 0 3),(7 0 7) 8.544(3 0 8),(7 0 6) 8.426 (1 0 4),(7 0 0) 8.544(3 0 8),(7 0 7) 8.426 (1 0 4),(7 0 1) 8.544(3 0 8),(7 0 8) 8.426 (1 0 4),(7 0 2) 8.544(3 0 8),(7 0 9) 8.426 (1 0 4),(7 0 3) 8.544(3 0 9),(7 0 0) 8.426 (1 0 4),(7 0 4) 8.544(3 0 9),(7 0 1) 8.426 (1 0 4),(7 0 5) 8.544(3 0 9),(7 0 2) 8.426 (1 0 4),(7 0 6) 8.544(3 0 9),(7 0 3) 8.426 (1 0 4),(7 0 7) 8.544(3 0 9),(7 0 4) 8.426 (1 0 5),(7 0 0) 8.544(3 0 9),(7 0 5) 8.426 (1 0 5),(7 0 1) 8.544(3 0 9),(7 0 6) 8.426 (1 0 5),(7 0 2) 8.544(3 0 9),(7 0 7) 8.426 (1 0 5),(7 0 3) 8.544(3 0 9),(7 0 8) 8.426 (1 0 5),(7 0 4) 8.544(3 0 9),(7 0 9) 8.426 (1 0 5),(7 0 5) 8.544

73 8 (1 0 0),(7 0 0) 8.544 (1 0 5),(7 0 6) 8.544(1 0 0),(7 0 1) 8.544 (1 0 5),(7 0 7) 8.544

(8.544) (1 0 0),(7 0 2) 8.544 (1 0 6),(7 0 0) 8.544(1 0 0),(7 0 3) 8.544 (1 0 6),(7 0 1) 8.544(1 0 0),(7 0 4) 8.544 (1 0 6),(7 0 2) 8.544(1 0 0),(7 0 5) 8.544 (1 0 6),(7 0 3) 8.544(1 0 0),(7 0 6) 8.544 (1 0 6),(7 0 4) 8.544(1 0 0),(7 0 7) 8.544 (1 0 6),(7 0 5) 8.544(1 0 1),(7 0 0) 8.544 (1 0 6),(7 0 6) 8.544(1 0 1),(7 0 1) 8.544 (1 0 6),(7 0 7) 8.544(1 0 1),(7 0 2) 8.544 (1 0 7),(7 0 0) 8.544(1 0 1),(7 0 3) 8.544 (1 0 7),(7 0 1) 8.544(1 0 1),(7 0 4) 8.544 (1 0 7),(7 0 2) 8.544(1 0 1),(7 0 5) 8.544 (1 0 7),(7 0 3) 8.544(1 0 1),(7 0 6) 8.544 (1 0 7),(7 0 4) 8.544(1 0 1),(7 0 7) 8.544 (1 0 7),(7 0 5) 8.544(1 0 2),(7 0 0) 8.544 (1 0 7),(7 0 6) 8.544(1 0 2),(7 0 1) 8.544 (1 0 7),(7 0 7) 8.544(1 0 2),(7 0 2) 8.544 (3 0 0),(5 0 0) 8.544(1 0 2),(7 0 3) 8.544 (3 0 0),(5 0 1) 8.544(1 0 2),(7 0 4) 8.544 (3 0 0),(5 0 2) 8.544(1 0 2),(7 0 5) 8.544 (3 0 0),(5 0 3) 8.544(1 0 2),(7 0 6) 8.544 (3 0 0),(5 0 4) 8.544(1 0 2),(7 0 7) 8.544 (3 0 0),(5 0 5) 8.544(1 0 3),(7 0 0) 8.544 (3 0 0),(5 0 6) 8.544(1 0 3),(7 0 1) 8.544 (3 0 0),(5 0 7) 8.544(1 0 3),(7 0 2) 8.544 (3 0 1),(5 0 0) 8.544(1 0 3),(7 0 3) 8.544 (3 0 1),(5 0 1) 8.544


53


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(3 0 1),(5 0 2) 8.544 (3 0 7),(5 0 0) 8.544(3 0 1),(5 0 3) 8.544 (3 0 7),(5 0 1) 8.544(3 0 1),(5 0 4) 8.544 (3 0 7),(5 0 2) 8.544(3 0 1),(5 0 5) 8.544 (3 0 7),(5 0 3) 8.544(3 0 1),(5 0 6) 8.544 (3 0 7),(5 0 4) 8.544(3 0 1),(5 0 7) 8.544 (3 0 7),(5 0 5) 8.544(3 0 2),(5 0 0) 8.544 (3 0 7),(5 0 6) 8.544(3 0 2),(5 0 1) 8.544 (3 0 7),(5 0 7) 8.544(3 0 2),(5 0 2) 8.544 79 6 (1 0 0),(5 0 0) 8.888(3 0 2),(5 0 3) 8.544 (1 0 0),(5 0 1) 8.888(3 0 2),(5 0 4) 8.544 ( 8.888) (1 0 0),(5 0 2) 8.888(3 0 2),(5 0 5) 8.544 (1 0 0),(5 0 3) 8.888(3 0 2),(5 0 6) 8.544 (1 0 0),(5 0 4) 8.888(3 0 2),(5 0 7) 8.544 (1 0 0),(5 0 5) 8.888(3 0 3),(5 0 0) 8.544 (1 0 1),(5 0 0) 8.888(3 0 3),(5 0 1) 8.544 (1 0 1),(5 0 1) 8.888(3 0 3),(5 0 2) 8.544 (1 0 1),(5 0 2) 8.888(3 0 3),(5 0 3) 8.544 (1 0 1),(5 0 3) 8.888(3 0 3),(5 0 4) 8.544 (1 0 1),(5 0 4) 8.888(3 0 3),(5 0 5) 8.544 (1 0 1),(5 0 5) 8.888(3 0 3),(5 0 6) 8.544 (1 0 2),(5 0 0) 8.888(3 0 3),(5 0 7) 8.544 (1 0 2),(5 0 1) 8.888(3 0 4),(5 0 0) 8.544 (1 0 2),(5 0 2) 8.888(3 0 4),(5 0 1) 8.544 (1 0 2),(5 0 3) 8.888(3 0 4),(5 0 2) 8.544 (1 0 2),(5 0 4) 8.888(3 0 4),(5 0 3) 8.544 (1 0 2),(5 0 5) 8.888(3 0 4),(5 0 4) 8.544 (1 0 3),(5 0 0) 8.888(3 0 4),(5 0 5) 8.544 (1 0 3),(5 0 1) 8.888(3 0 4),(5 0 6) 8.544 (1 0 3),(5 0 2) 8.888(3 0 4),(5 0 7) 8.544 (1 0 3),(5 0 3) 8.888(3 0 5),(5 0 0) 8.544 (1 0 3),(5 0 4) 8.888(3 0 5),(5 0 1) 8.544 (1 0 3),(5 0 5) 8.888(3 0 5),(5 0 2) 8.544 (1 0 4),(5 0 0) 8.888(3 0 5),(5 0 3) 8.544 (1 0 4),(5 0 1) 8.888(3 0 5),(5 0 4) 8.544 (1 0 4),(5 0 2) 8.888(3 0 5),(5 0 5) 8.544 (1 0 4),(5 0 3) 8.888(3 0 5),(5 0 6) 8.544 (1 0 4),(5 0 4) 8.888(3 0 5),(5 0 7) 8.544 (1 0 4),(5 0 5) 8.888(3 0 6),(5 0 0) 8.544 (1 0 5),(5 0 0) 8.888(3 0 6),(5 0 1) 8.544 (1 0 5),(5 0 1) 8.888(3 0 6),(5 0 2) 8.544 (1 0 5),(5 0 2) 8.888(3 0 6),(5 0 3) 8.544 (1 0 5),(5 0 3) 8.888(3 0 6),(5 0 4) 8.544 (1 0 5),(5 0 4) 8.888(3 0 6),(5 0 5) 8.544 (1 0 5),(5 0 5) 8.888(3 0 6),(5 0 6) 8.544 89 8 (1 0 0),(7 0 0) 9.434(3 0 6),(5 0 7) 8.544 (9.434) (1 0 0),(7 0 1) 9.434


54


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(1 0 0),(7 0 2) 9.434 (1 0 6),(7 0 0) 9.434(1 0 0),(7 0 3) 9.434 (1 0 6),(7 0 1) 9.434(1 0 0),(7 0 4) 9.434 (1 0 6),(7 0 2) 9.434(1 0 0),(7 0 5) 9.434 (1 0 6),(7 0 3) 9.434(1 0 0),(7 0 6) 9.434 (1 0 6),(7 0 4) 9.434(1 0 0),(7 0 7) 9.434 (1 0 6),(7 0 5) 9.434(1 0 1),(7 0 0) 9.434 (1 0 6),(7 0 6) 9.434(1 0 1),(7 0 1) 9.434 (1 0 6),(7 0 7) 9.434(1 0 1),(7 0 2) 9.434 (1 0 7),(7 0 0) 9.434(1 0 1),(7 0 3) 9.434 (1 0 7),(7 0 1) 9.434(1 0 1),(7 0 4) 9.434 (1 0 7),(7 0 2) 9.434(1 0 1),(7 0 5) 9.434 (1 0 7),(7 0 3) 9.434(1 0 1),(7 0 6) 9.434 (1 0 7),(7 0 4) 9.434(1 0 1),(7 0 7) 9.434 (1 0 7),(7 0 5) 9.434(1 0 2),(7 0 0) 9.434 (1 0 7),(7 0 6) 9.434(1 0 2),(7 0 1) 9.434 (1 0 7),(7 0 7) 9.434(1 0 2),(7 0 2) 9.434 (3 0 0),(5 0 0) 9.434(1 0 2),(7 0 3) 9.434 (3 0 0),(5 0 1) 9.434(1 0 2),(7 0 4) 9.434 (3 0 0),(5 0 2) 9.434(1 0 2),(7 0 5) 9.434 (3 0 0),(5 0 3) 9.434(1 0 2),(7 0 6) 9.434 (3 0 0),(5 0 4) 9.434(1 0 2),(7 0 7) 9.434 (3 0 0),(5 0 5) 9.434(1 0 3),(7 0 0) 9.434 (3 0 0),(5 0 6) 9.434(1 0 3),(7 0 1) 9.434 (3 0 0),(5 0 7) 9.434(1 0 3),(7 0 2) 9.434 (3 0 1),(5 0 0) 9.434(1 0 3),(7 0 3) 9.434 (3 0 1),(5 0 1) 9.434(1 0 3),(7 0 4) 9.434 (3 0 1),(5 0 2) 9.434(1 0 3),(7 0 5) 9.434 (3 0 1),(5 0 3) 9.434(1 0 3),(7 0 6) 9.434 (3 0 1),(5 0 4) 9.434(1 0 3),(7 0 7) 9.434 (3 0 1),(5 0 5) 9.434(1 0 4),(7 0 0) 9.434 (3 0 1),(5 0 6) 9.434(1 0 4),(7 0 1) 9.434 (3 0 1),(5 0 7) 9.434(1 0 4),(7 0 2) 9.434 (3 0 2),(5 0 0) 9.434(1 0 4),(7 0 3) 9.434 (3 0 2),(5 0 1) 9.434(1 0 4),(7 0 4) 9.434 (3 0 2),(5 0 2) 9.434(1 0 4),(7 0 5) 9.434 (3 0 2),(5 0 3) 9.434(1 0 4),(7 0 6) 9.434 (3 0 2),(5 0 4) 9.434(1 0 4),(7 0 7) 9.434 (3 0 2),(5 0 5) 9.434(1 0 5),(7 0 0) 9.434 (3 0 2),(5 0 6) 9.434(1 0 5),(7 0 1) 9.434 (3 0 2),(5 0 7) 9.434(1 0 5),(7 0 2) 9.434 (3 0 3),(5 0 0) 9.434(1 0 5),(7 0 3) 9.434 (3 0 3),(5 0 1) 9.434(1 0 5),(7 0 4) 9.434 (3 0 3),(5 0 2) 9.434(1 0 5),(7 0 5) 9.434 (3 0 3),(5 0 3) 9.434(1 0 5),(7 0 6) 9.434 (3 0 3),(5 0 4) 9.434(1 0 5),(7 0 7) 9.434 (3 0 3),(5 0 5) 9.434


55


p) q (k, k1, k2), (k′, k′

1, k′2) Max p(

√p) q (k, k1, k2), (k

′, k′1, k

′2) Max

(3 0 3),(5 0 6) 9.434(3 0 3),(5 0 7) 9.434(3 0 4),(5 0 0) 9.434(3 0 4),(5 0 1) 9.434(3 0 4),(5 0 2) 9.434(3 0 4),(5 0 3) 9.434(3 0 4),(5 0 4) 9.434(3 0 4),(5 0 5) 9.434(3 0 4),(5 0 6) 9.434(3 0 4),(5 0 7) 9.434(3 0 5),(5 0 0) 9.434(3 0 5),(5 0 1) 9.434(3 0 5),(5 0 2) 9.434(3 0 5),(5 0 3) 9.434(3 0 5),(5 0 4) 9.434(3 0 5),(5 0 5) 9.434(3 0 5),(5 0 6) 9.434(3 0 5),(5 0 7) 9.434(3 0 6),(5 0 0) 9.434(3 0 6),(5 0 1) 9.434(3 0 6),(5 0 2) 9.434(3 0 6),(5 0 3) 9.434(3 0 6),(5 0 4) 9.434(3 0 6),(5 0 5) 9.434(3 0 6),(5 0 6) 9.434(3 0 6),(5 0 7) 9.434(3 0 7),(5 0 0) 9.434(3 0 7),(5 0 1) 9.434(3 0 7),(5 0 2) 9.434(3 0 7),(5 0 3) 9.434(3 0 7),(5 0 4) 9.434(3 0 7),(5 0 5) 9.434(3 0 7),(5 0 6) 9.434(3 0 7),(5 0 7) 9.434

56

Table 5.3: Crosscorrelations of D(M(s(n))) for p ≤ 97.

p q k k′ maxd,d′ Max

τ(Ca,b)(τ) min

d,d′ Maxτ

(Ca,b)(τ)√

p + 2

11 5 1 1 11 9.21 5.3171 2 5.213 4.3691 3 5.314 4.6751 4 5.213 3.8922 2 11 9.212 3 5.314 3.9332 4 5.314 4.6753 3 11 9.213 4 5.213 4.3694 4 11 9.21

13 3 1 1 13 11.533 5.6061 2 5.568 4.3592 2 13 11.533

4 1 1 13 111 3 5.385 3.6063 3 13 11

6 1 1 13 111 5 5.568 3.6065 5 13 11

17 4 1 1 17 15 6.1231 3 6.083 4.1233 3 17 15

8 1 1 17 151 3 6.083 5.5061 5 6.123 5.5371 7 6.123 4.1233 3 17 153 5 6.123 4.1233 7 6.123 5.5375 5 17 155 7 6.083 5.5067 7 17 15

19 3 1 1 19 17.521 6.3591 2 6.083 5.2922 2 19 17.521

6 1 1 19 171 5 6.245 4.3595 5 19 17

9 1 1 19 17.0641 2 6.314 5.7521 4 6.314 5.3921 5 6.34 5.7421 7 6.282 5.3591 8 6.314 4.6962 2 19 17.064


57

continued from previous pagep q k k′ max

d,d′ Maxτ

(Ca,b)(τ) mind,d′ Max

τ(Ca,b)(τ)

√p + 2

2 4 6.282 5.6512 5 6.34 5.4782 7 6.282 4.6882 8 6.282 5.3594 4 19 17.0644 5 6.34 4.7024 7 6.34 5.4784 8 6.34 5.7425 5 19 17.0645 7 6.282 5.6515 8 6.314 5.3927 7 19 17.0647 8 6.314 5.7528 8 19 17.064

29 4 1 1 29 27 7.3851 3 7.28 5.3853 3 29 27

7 1 1 29 27.1031 2 7.307 6.5141 3 7.384 6.6311 4 7.291 6.5551 5 7.384 6.6311 6 7.307 5.8272 2 29 27.1032 3 7.384 6.6312 4 7.364 6.5862 5 7.364 5.8142 6 7.384 6.6313 3 29 27.1033 4 7.291 5.8293 5 7.364 6.5863 6 7.291 6.5554 4 29 27.1034 5 7.384 6.6314 6 7.384 6.6315 5 29 27.1035 6 7.307 6.5146 6 29 27.103

31 3 1 1 31 29.513 7.5681 2 7 6.5572 2 31 29.513

5 1 1 31 29.1971 2 7.492 6.641 3 7.562 6.8691 4 7.492 6.1572 2 31 29.197


58


d,d′ Maxτ


τ(Ca,b)(τ)

√p + 2

2 3 7.562 6.1842 4 7.562 6.8693 3 31 29.1973 4 7.492 6.644 4 31 29.197

6 1 1 31 291 5 7.55 5.5685 5 31 29

10 1 1 31 291 3 7.562 6.9691 7 7.499 7.1211 9 7.562 5.5683 3 31 293 7 7.499 5.5683 9 7.499 7.1217 7 31 297 9 7.562 6.9699 9 31 29

37 3 1 1 37 35.511 8.0831 2 7.81 72 2 37 35.511

4 1 1 37 351 3 8.062 6.0833 3 37 35

6 1 1 37 351 5 7.937 6.0835 5 37 35

9 1 1 37 35.0621 2 8.081 7.2811 4 8.081 7.2811 5 8.07 7.3771 7 8.028 7.081 8 8.081 6.432 2 37 35.0622 4 8.028 7.4012 5 8.07 7.2092 7 8.028 6.4182 8 8.028 7.084 4 37 35.0624 5 8.07 6.4274 7 8.07 7.2094 8 8.07 7.3775 5 37 35.0625 7 8.028 7.4015 8 8.081 7.2817 7 37 35.062


59


d,d′ Maxτ


τ(Ca,b)(τ)

√p + 2

7 8 8.081 7.2818 8 37 35.062

41 4 1 1 41 39 8.4031 3 8.062 6.4033 3 41 39

5 1 1 41 39.1951 2 8.402 7.7421 3 8.363 7.5621 4 8.402 7.0212 2 41 39.1952 3 8.363 7.0062 4 8.363 7.5623 3 41 39.1953 4 8.402 7.7424 4 41 39.195

8 1 1 41 391 3 8.394 7.811 5 8.336 7.6811 7 8.336 6.4033 3 41 393 5 8.336 6.4033 7 8.336 7.6815 5 41 395 7 8.394 7.817 7 41 39

10 1 1 41 391 3 8.363 7.9191 7 8.402 7.7211 9 8.363 6.4033 3 41 393 7 8.402 6.4033 9 8.402 7.7217 7 41 397 9 8.363 7.9199 9 41 39

43 3 1 1 43 41.509 8.5571 2 8.544 7.2112 2 43 41.509

6 1 1 43 411 5 8.544 6.5575 5 43 41

7 1 1 43 41.1011 2 8.484 7.6341 3 8.426 7.7151 4 8.537 7.7321 5 8.426 7.715


60


d,d′ Maxτ


τ(Ca,b)(τ)

√p + 2

1 6 8.484 6.9982 2 43 41.1012 3 8.426 7.7152 4 8.502 7.6022 5 8.502 6.9962 6 8.426 7.7153 3 43 41.1013 4 8.537 6.9863 5 8.502 7.6023 6 8.537 7.7324 4 43 41.1014 5 8.426 7.7154 6 8.426 7.7155 5 43 41.1015 6 8.484 7.6346 6 43 41.101

53 4 1 1 53 51 9.281 3 9.22 7.283 3 53 51

61 3 1 1 61 59.506 9.811 2 9.539 8.7182 2 61 59.506

4 1 1 61 591 3 9.434 7.813 3 61 59

5 1 1 61 59.1941 2 9.804 9.0941 3 9.518 8.4771 4 9.804 8.4262 2 61 59.1942 3 9.518 8.3242 4 9.518 8.4773 3 61 59.1943 4 9.804 9.0944 4 61 59.194

6 1 1 61 591 5 9.644 7.815 5 61 59

10 1 1 61 591 3 9.81 9.0831 7 9.804 9.171 9 9.81 7.813 3 61 593 7 9.804 7.813 9 9.804 9.177 7 61 59


61


d,d′ Maxτ


τ(Ca,b)(τ)

√p + 2

7 9 9.81 9.0839 9 61 59

67 3 1 1 67 65.506 10.1851 2 10.149 8.8882 2 67 65.506

6 1 1 67 651 5 10.149 8.1855 5 67 65

71 5 1 1 71 69.193 10.4261 2 10.186 9.171 3 10.404 9.6291 4 10.186 8.9592 2 71 69.1932 3 10.404 9.0362 4 10.404 9.6293 3 71 69.1933 4 10.186 9.174 4 71 69.193

7 1 1 71 69.11 2 10.393 9.5021 3 10.413 9.6641 4 10.32 9.5431 5 10.413 9.6641 6 10.393 8.8582 2 71 69.12 3 10.413 9.6642 4 10.425 9.7562 5 10.425 8.8352 6 10.413 9.6643 3 71 69.13 4 10.32 8.873 5 10.425 9.7563 6 10.32 9.5434 4 71 69.14 5 10.413 9.6644 6 10.413 9.6645 5 71 69.15 6 10.393 9.5026 6 71 69.1

10 1 1 71 691 3 10.404 9.8751 7 10.421 9.7711 9 10.404 8.4263 3 71 693 7 10.421 8.4263 9 10.421 9.771


62


d,d′ Maxτ


τ(Ca,b)(τ)

√p + 2

7 7 71 697 9 10.404 9.8759 9 71 69

73 3 1 1 73 71.505 10.5441 2 9.849 9.5392 2 73 71.505

4 1 1 73 711 3 10.44 8.5443 3 73 71

6 1 1 73 711 5 10.536 8.5445 5 73 71

8 1 1 73 711 3 10.44 9.8811 5 10.533 9.9081 7 10.533 8.5443 3 73 713 5 10.533 8.5443 7 10.533 9.9085 5 73 715 7 10.44 9.8817 7 73 71

9 1 1 73 71.0611 2 10.505 9.8891 4 10.505 9.5531 5 10.542 9.7291 7 10.515 9.5861 8 10.505 8.8832 2 73 71.0612 4 10.515 9.8752 5 10.542 9.7292 7 10.515 8.8852 8 10.515 9.5864 4 73 71.0614 5 10.542 8.8914 7 10.542 9.7294 8 10.542 9.7295 5 73 71.0615 7 10.515 9.8755 8 10.505 9.5537 7 73 71.0617 8 10.505 9.8898 8 73 71.061

79 3 1 1 79 77.505 10.8881 2 10.44 9.8492 2 79 77.505


63


d,d′ Maxτ


τ(Ca,b)(τ)

√p + 2

6 1 1 79 771 5 10.817 8.8885 5 79 77

89 4 1 1 89 87 11.4341 3 11.18 9.4343 3 89 87

8 1 1 89 871 3 11.392 10.8171 5 11.358 10.7081 7 11.358 9.4343 3 89 873 5 11.358 9.4343 7 11.358 10.7085 5 89 875 7 11.392 10.8177 7 89 87

97 3 1 1 97 95.504 11.8491 2 11.79 10.5832 2 97 95.504

4 1 1 97 951 3 11.705 9.8493 3 97 95

6 1 1 97 951 5 11.79 9.8495 5 97 95

8 1 1 97 951 3 11.738 11.181 5 11.796 11.0911 7 11.796 9.8493 3 97 953 5 11.796 9.8493 7 11.796 11.0915 5 97 955 7 11.738 11.187 7 97 95

64

Bibliography

[1] S. W. Golomb, Shift Register Sequences, revised ed. P.O.Box 2837: Aegean Park

Press, 1982, with portions co-authored by Lloyd R. Welch, Richard M. Goldstein,

and Alfred W. Hales.

[2] M. D. S. D. H. Green and N. Marzoukos, “Linear complexity of polyphase power

residue sequences,” IEE Proc.-Commun., vol. 149, no. 4, pp. 195–201, 2002.

[3] D. H. Green and P. R. GReen, “Polyphase related-prime sequences,” IEE Proc.-

Comput. Digit. Tech, vol. 148, no. 2, pp. 53–62, 2001.

[4] T. Helleseth and P. V. Kumar, “Sequences with low correlation,” in Handbook of

Coding Theory, V. S. Pless and W. C. Huffman, Eds. Elsevier Science, 1998,

ch. 21, pp. 1765–1853.

[5] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum

Communications Handbook, revised ed. McGraw-Hill, Inc., 1994.

[6] T. Kasami, “Weight distribution formula for some class of cyclic codes,” Cordi-

nated Science Lab., Univ. Illinois, Urbana, Tech. Report R-108 (AD 632574), 1966.

65

[7] R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” vol. IT-13,

pp. 619–621, Oct. 1967.

[8] R. J. Turyn, “The linear generation of the legendre sequendes,” SIAM J. Appl. Math,

vol. 12, no. 1.

[9] H.-Y. Song and S. W. Golomb, “On the existence of cyclic Hadamard difference

sets,” vol. IT–40, no. 4, pp. 1266–1268, July 1994.

[10] S. W. Golomb and H.-Y. Song, “A conjecture on the existence of cyclic hadamard

difference sets,” Journal of Statistical Planning and Inference, vol. 62, pp. 39–41,

1997.

[11] T. H. C. Ding and W. Shan, “On the linear complexity of legendre sequences,”

IEEE Trans. Inform. Theory, vol. 44, no. 3, pp. 1276–1278, May 1998.

[12] J.-H. Kim and H.-Y. Song, “Trace representation of legendre sequences,” Designs,

Codes and Cryptography, vol. 24, pp. 343–348, 2001.

66

²DG ë�H ¹כ ��

��©� "4� eç#�ÀÓ Ãº\P�_� �©� ñ�©��'a :£¤$í\� �'aô�Ç ��½̈

�©��'a :£¤$ís� ÄºÃºô�Ç Ãº\P��̀¦ ¹1Ô��H ë�H]j��H YUs�� r�Û¼%7�s�� CDMA :�x��

�̀¦ q�2�©ô�Ç @/%i� SX�íß� :�x�� r�Û¼%7�\�"f Õª $í0px�̀¦ �¾Ó�©�r�v�l� 0Aô�Ç ×�æ¹כ õ�

]j ×�æ ��s��. Ãº\P�_� �©��'a :£¤$í�Ér ß¼>� ��l� �©��'a :£¤$íõ� �©� ñ �©��'a :£¤

$íÜ¼�Ð ��*'��HX<, 6£x6 x÷&��H r�Û¼%7�_� $í��\� �� s� ¿º��t� :£¤$ís� �̧¿º

l��̧&÷̈½¹כ ��¦ ô�Ç��t� :£¤$íëß�s� ¹כ��9�l��̧ ��. s�� ØÔ�©�×¼ØÔ Ãº\P�s�

þj&h�_� ��l� �©��'a :£¤$í�̀¦ °ú��H ��õכ °ú s� ��©� "4� eç#�ÀÓ Ãº\P� ¢̧ô�Ç ÄºÃº

ô�Ç ��l��©��'a :£¤$í�̀¦ °ú��H��. �:r �7Hë�H\�"f��H "é¶r��H�̀¦ ��ÀDKÜ¼�Ð+� ��©� "4�

eç#�ÀÓ Ãº\P�[þt_� |9�½+Ë�̀¦ Òqt$í ��¦, ÅÒ#Q�� q-�©� eç#�ÀÓ Ãº\P�\� ��ª�ô�Ç ��

8̈��̀¦ &h�6 x�<ÊÜ¼�Ð+� q-�©� Ãº\P�[þt�̀¦ Òqt$í ��¦, s�[þt_� �©� ñ �©��'a°ú̀�כ¦ >�íß�ô�Ç

��. #�l�"f ��8̈��Ér �©�ÃºC�(the constant multiple), Ä»�� 8̈� s�1lx(the affine

shift), í�H8̈� s�1lx(the cyclic shift), Ãº\P��̀¦ {9�&ñçß��m�� (̀#Q"f {9��H ��8̈�(the

decimation)1px�̀¦ �¦�9 ô�Ç��. ÅÒ#Q�� q-�©� eç#�ÀÓ Ãº\P�\� �©�ÃºC�\�¦ �#� ëß�

��H Ãº\P�_� |9�½+Ëõ� "é¶r��H�̀¦ ��ÀDKÜ¼�Ð+� Òqt$í÷&��H Ãº\P�_� |9�½+Ës� 1lx{9� ��.

s� |9�½+Ë_� "f�Ð ��Ér ¿º "é¶�è Ãº\P� ��s�_� �©� ñ�©��'a :£¤$í�̀¦ >�íß�ô�Ç ��õ� �©�

ñ�©��'a°úכ_� �©�ô�Çs�√

p + 2e��̀¦ ]jîß�ô�Ç��. ��6£§Ü¼�Ð Ä»�� 8̈� s�1lxõ� �©�

ÃºC� ��8̈��̀¦ 1lxr�\� &h�6 x �#� Ãº\P�_� |9�½+Ë�̀¦ Òqt$íô�Ç��. 2>h_� �©��'a°úכëß��̀¦

°ú��H Ãº\P��©��̀¦ �̧��"f �'a¹1Ïô�Ç ��õ� s�[þt �©��'a°úכ_� þj@/°úכs� &ñSX�y�√

ps�

67

��. ��t�}��Ü¼�Ð Ãº\P��̀¦ {9�&ñçß��m�� (̀#Q"f {9��H ��8̈�(the decimation)õ� �©�

ÃºC�\�¦ ��H ��8̈��̀¦ &h�6 xô�Ç��. q-�©� eç#�ÀÓ Ãº\P�\� {9�&ñ çß��m�� (̀#Q"f {9�

��H ��8̈��̀¦ &h�6 x �%i��̀¦M: �©� ñ�©��'a°úכs�√

p�Ð�� B�Äº ß¼�¦ d��t�#Q p − 2�Ð

��̧ ß¼�� °ú ��. Õª�Q�� ©�ÃºC��8̈�õ� {9�&ñ çß��m�� (̀#Q"f {9��H ��8̈��̀¦

1lxr�\� &h�6 x �%i��̀¦M:��H <Éªp��Ðî�r &h�s� �'a¹1Ï�)a��. �©�ÃºC� ��8̈�_� �'a&h�\�"f

"f�Ð ��Ér ¿º>h_� Ãº\P�\� @/K� ��_� Ãº\P��̀¦ {9�&ñ çß��m�� (̀#Q"f {9��H ��

8̈�(decimation)Ü¼�Ð ÂÒ'� ��:r Ãº\P�õ� ��Ér Ãº\P��̀¦ {9�&ñ çß��m�� (̀#Q"f {9�

��H ��8̈�(decimation)Ü¼�Ð ÂÒ'� ��:r Ãº\P��s�_� �©� ñ�©��'a°úכ�Ér Ãº\P��̀¦ {9�&ñ

çß��m�� (̀#Q"f {9��H ��8̈�(decimation)\� �©��'a\O�s�√

p + 2\� �©�ô�Ç�)a��.

Ùþ�d��÷&��H ú́�: ØÔ�©�×¼ØÔ Ãº\P�, ��©� "4� eç#�ÀÓ Ãº\P�, ��l��©��'a:£¤$í, �©� ñ�©�

�'a:£¤$í

68

on the crosscorrelation of polyphase power residue...

Documents