2010 nova science publishers, inc. proof › ~jdebeule › book › c8.pdf · in: current research...

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In: Current research topics in Galois geometryEditor: J. De Beule, pp. 141-168

ISBN 0000000000c© 2010 Nova Science Publishers, Inc.

Chapter 81

GALOIS GEOMETRIES AND CODING THEORY2

Ivan Landjev∗†and Leo Storme‡§3

Abstract4

Many problems on linear codes can be retranslated into equivalent problems on5

specific substructures in Galois geometries. This implies that geometrical methods can6

be used to investigate problems on linear codes, and vice versa that coding-theoretical7

methods can be used to investigate problems in Galois geometries. We present in this8

article a number of the most interesting links between linear codes and substructures9

in Galois geometries. We start with some basic facts from coding theory to make the10

article self-contained. Then we present the important links between n-arcs in Galois11

geometries and linear MDS codes, minihypers and linear codes meeting the Griesmer12

bound, links between the extendability of linear codes and blocking sets, saturating13

sets and the covering radius of linear codes, and conclude with the linear codes aris-14

ing from the incidence matrices of Galois geometries, illustrating their relevance for15

Galois geometries by giving an upper bound on the sizes of sets of points in PG(N,q)16

having in each of their points a tangent hyperplane.17

Key Words: Linear codes, Arcs, MDS codes, Minihypers, Griesmer bound, Saturating sets,18

Covering radius, Extendability, Incidence Matrices19

AMS Subject Classification: 05B25, 51E15, 51E20, 51E21, 51E22, 94B05.20

∗New Bulgarian University, 21 Montevideo str., 1618 Sofia, Bulgaria, and Institute of Mathematics andInformatics, BAS, 8 Acad. G. Bonchev str., 1113, Sofia, Bulgaria. E-mail: [email protected]

†This research was supported by the Strategic Development Fund of the New Bulgarian University underContract 357/14.05.2009.

‡Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281-S22, 9000Ghent, Belgium. E-mail: [email protected]

§This research was supported by the project Combined algorithmic and theoretical study of combinatorialstructures between the Research Foundation – Flanders (FWO) and the Bulgarian Academy of Sciences. Thisresearch also takes place within the project Linear codes and cryptography of the Research Foundation –Flanders (FWO) (project nr. G.0317.06)

[email protected]

[email protected]

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142 Ivan Landjev and Leo Storme

1 Linear codes over finite fields1

1.1 General definitions2

Let Fnq denote the vector space of all n-tuples over the q-element field Fq. Every k-3

dimensional subspace C of Fnq is called a q-ary linear code C of length n and dimension k,4

or an [n,k]q code [57]. The inner product of the vectors u = (u1, . . . ,un) and v = (v1, . . . ,vn)5

from Fnq is defined by6

u · v = u1v1 + · · ·+unvn.7

Two vectors are said to be orthogonal if their inner product is 0. The set of all vectors of Fnq8

orthogonal to all codewords from C is called the dual code C⊥ of C:9

C⊥ = {x ∈ Fnq|x · y = 0 for all y ∈C}.10

Clearly, the code C⊥ is a linear [n,n− k]q code.11

A k-by-n matrix G having as rows the vectors of a basis of C is called a generator matrix12

of C. A generator matrix H of the code C⊥, dual to C, is a parity check matrix for C.13

The number of non-zero positions in a vector x ∈ Fnq is called the Hamming weight w(x)14

of x. The Hamming distance d(x,y) between two vectors x,y ∈ Fnq is defined by15

d(x,y) = w(x− y).16

The minimum distance of a linear code C is17

d(C) = min{d(x,y)|x,y ∈C,x 6= y}= min{w(c)|c ∈C,c 6= 0}.18

A q-ary linear code of length n, dimension k, and minimum distance d, is referred to as19

an [n,k,d]q or [n,k,d] code.20

Theorem 1.1. Let C be a linear code over Fq with parity check matrix H. If any δ− 121

columns of H are linearly independent over Fq, then d(C)≥ δ. The minimum distance of C22

is d if and only if any d−1 columns of H are linearly independent over Fq and there exist23

d linearly dependent columns in H.24

A central problem in coding theory is to optimize one of the parameters n,k, or d of a25

linear code, given the other two. This leads to the following three optimization problems:26

(A) Find nq(k,d), the smallest value of n for which there exists an [n,k,d]q code.27

(B) Find Kq(n,d), the largest value of k for which there exists an [n,k,d]q code.28

(C) Find Dq(n,k), the largest value of d for which there exists an [n,k,d]q code.29

If we know the exact values of one of the functions defined in (A)–(C) for all pairs of30

arguments, we can find the exact values of the remaining two functions.31

A code of length nq(k,d), dimension k, and minimum distance d, is said to be optimal32

with respect to n. Similarly, codes with parameters [n,Kq(n,d),d]q and [n,k,Dq(n,k)]q33

are called optimal with respect to k and d. It may turn out that a code, which is optimal34

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Galois geometries and coding theory 143

with respect to one of the parameters n, k, d, is not optimal with respect to (one of) the1

other two parameters. However, a code which is optimal with respect to the length is also2

optimal with respect to the dimension and the minimum distance. This follows by the easy3

observation that the function nq(k,d) is strictly increasing in both of its arguments, i.e.4

nq(k + 1,d) > nq(k,d) and nq(k,d + 1) > nq(k,d). Hence, a code which minimizes n for5

given k and d, maximizes k for given n and d, and at the same time maximizes d for given6

n and k. Thus the function defined in (A) is the most sensitive of all three. In Section 4, we7

present a natural lower bound on nq(k,d), the so-called Griesmer bound.8

1.2 Automorphisms of linear codes9

Let C1 and C2 be two linear [n,k,d]q codes. They are said to be semi-linearly equivalent if10

the codewords of C2 can be obtained from the codewords of C1 via a sequence of transfor-11

mations of the following types:12

(i) permutation on the set of coordinate positions;13

(ii) multiplication of the elements in a given position by a non-zero element of Fq;14

(iii) application of a field automorphism to the elements in all coordinate positions.15

1.3 The spectrum of a linear code16

Given an [n,k,d]q code C, we denote by Ai the number of codewords of weight i in C.17

The sequence of integers (A0, . . . ,An, . . .) is called the spectrum of C. Sometimes, it is18

convenient to work with the so-called Hamming weight enumerator of C defined by19

WC(X ,Y ) =n

∑i=0

AiXn−iY i.20

1.4 Generalized Hamming weights21

Let C be a linear [n,k]q code. The set supp(C) of those coordinate positions, where not all22

the codewords of C are zero, is called the support of C. The support of a codeword is the23

support of the one-dimensional subcode generated by this codeword. The r-th generalized24

Hamming weight dr(C) is defined to be the cardinality of the minimal support of an [n,r]q25

subcode of C, 1 ≤ r ≤ k, i.e.,26

dr(C) = min{|supp(D)|

∣∣ D is an [n,r]q subcode of C}

.27

Obviously, d1(C) is the minimum distance of C. The following theorems give some funda-28

mental properties of the generalized Hamming weights.29

Theorem 1.2. (Wei [74]) For every linear [n,k]q code C,30

0 < d1(C) < d2(C) < · · · < dk(C) ≤ n.31

Theorem 1.3. (Wei [74]) Let H be a parity check matrix of the linear code C, then dr(C) = δ32

if and only if33

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(a) any δ−1 columns of H have rank larger than or equal to δ− r;1

(b) there exist δ columns in H of rank δ− r.2

Theorem 1.4. (Wei [74]) Let C be a linear [n,k]q code and let C⊥ be its dual code, then3

{dr(C) |r = 1, . . . ,k}∪{n+1−dr(C⊥) |r = 1, . . . ,n− k} = {1,2, . . . ,n}.4

Theorem 1.5 (The generalized Singleton bound). (Wei [74])5

dr(C) ≤ n− k + r, r = 1, . . . ,k.6

2 Arcs in Galois geometries7

?〈chap:arcs〉?2.1 Multiarcs and minihypers8

Let P be the set of points of the projective geometry PG(N,q). Every mapping K : P → N9

from the points of PG(N,q) to the non-negative integers is called a multiset in PG(N,q).10

This mapping is extended in a natural way to the subsets Q of P by K (Q ) = ∑P∈Q K (P).11

The integer K (P) is called the multiplicity of the point P and n = ∑P∈P K (P) is called the12

cardinality of K . The support supp(K ) of a multiset K is the set of all points of positive13

multiplicity. A multiset is said to be projective if K (P) ∈ {0,1} for all points P. Projective14

multisets can be considered as sets of points by identifying them with their support.15

Given a finite set Q of points in PG(N,q), we define the characteristic multiset χQ by:16

χQ (P) ={

1 if P ∈ Q ,0 if P 6∈ Q .

17

A multiset in PG(N,q) is called an (n,w;N,q)-multiarc or (n,w;N,q)-arc if18

(a) K (P ) = n;19

(b) K (H)≤ w for any hyperplane H, and there exists a hyperplane H0 with K (H0) = w.20

A multiset in PG(N,q) is called an (n,w;N,q)-blocking multiset or (n,w;N,q)-21

minihyper if22

(a) K (P ) = n;23

(b) K (H)≥ w for any hyperplane H, and there exists a hyperplane H0 with K (H0) = w.24

We will speak of (n,w)-multiarcs or (n,w)-minihypers if the geometry PG(N,q) we25

consider is clear from the context.26

The characteristic function of a subspace of dimension u in PG(N,q), u ≤ N, is a pro-27

jective minihyper with parameters (vu+1,vu), where vN = qN−1q−1 .28

2.2 Equivalence of multisets29

Two multisets K in PG(N,q) and K ′ in PG(N′,q′) are said to be equivalent if there exists30

a collineation ψ : 〈supp(K )〉 → 〈supp(K ′)〉, such that K (P) = K ′(ψ(P)) for every point31

P ∈ 〈supp(K )〉. Here 〈Q 〉, where Q ⊆ P is the subspace of PG(N,q) generated by the32

points of Q .33

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2.3 Arcs and codes1

There exists a familiar correspondence between the linear codes of full length (i.e. codes in2

which no coordinate position is identically zero) and the multiarcs in the projective geome-3

tries PG(N,q). Let C be an [n,k]q linear code of full length and let4

G = [c1, . . . ,ck]t = [g1, . . . ,gn], ci ∈ Fnq, gi ∈ Fk

q,5

be a generator matrix of C. We define the multiarc KS induced by the sequence of codewords6

S = [c1, . . . ,ck] of C by7

KS : P = PG(k−1,q)→ N : P 7→∣∣{ j | P = λ jg j, for some λ j ∈ Fq \{0}

}∣∣ .8

The code C and the multiarc KS are said to be associated to each other. A multiarc associ-9

ated with an [n,k,d]q code has parameters (n,n−d;k−1,q). Clearly, a linear code can be10

associated to different arcs, but we have the following theorem.11

〈thm:DS〉Theorem 2.1. For every multiset K of cardinality n in PG(k− 1,q), there exists a linear12

code C of full length in Fnq and a generating sequence S of C that induce K . Two multiarcs13

K1 and K2 in PG(k− 1,q) associated with the linear codes C1 and C2, respectively, are14

equivalent if and only if the codes C1 and C2 are semi-linearly equivalent.15

Theorem 2.1 can be further generalized for linear codes over finite chain rings and arcs16

in projective Hjelmslev geometries (see e.g. [52]).17

Let C be a linear code and let K be a multiarc associated with C. Denote by s the18

maximal multiplicity of a point from P . The minihyper F = sχP −K is called a minihy-19

per associated with C (respectively, a minihyper associated with K ). Note that different20

multiarcs can be associated with the same minihyper. Since21

F = sχP −K = (s+a)χP − (K +aχP ), a ∈ N,22

the multiarcs K and K ′ = K + aχP give rise to the same minihyper. Conversely, if K23

and K ′ are two multiarcs that give rise to the same minihyper, then K ′−K = aχP , a ∈ Z.24

Minihypers will be studied in more detail in Section 4, in relation with the problem of linear25

codes meeting the Griesmer bound.26

Given an (n,w;k−1,q)-arc, denote by ai the number of hyperplanes H with K (H) = i.27

The sequence (ai)i≥0 is called the spectrum of K . If C is a linear code associated with K28

with spectrum (Ai)i≥0, then ai = An−i/(q−1) for i = 0, . . . ,n.29

2.4 Weight hierarchy and generalized spectra for arcs30

Given an (n,w)-arc K in PG(N,q), we define wr as the maximal multiplicity of an r-31

dimensional subspace of PG(N,q):32

wr = wr(K ) = max∆

K (∆),33

where ∆ runs over all r-dimensional subspaces of PG(N,q). By definition, wN−1 = w and34

wN = n.35

For the numbers wr, we have the following straightforward result.36

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Theorem 2.2. Let K be a non-degenerate (n,w)-arc in PG(N,q) (i.e. an arc with1

〈supp(K )〉= PG(N,q)), then2

0 < w0 < w1 < · · ·< wN−1 < wN = n.3

The ordered (N +1)-tuple (w0,w1, . . . ,wN−1,wN) is called the weight hierarchy of K .4

2.5 Constructions for arcs5

〈subs:constrarcs〉Sum of multisets6

Let K1 and K2 be multiarcs in PG(N,q) with parameters (n1,w1) and (n2,w2), and let7

a,b ∈ Q be rational numbers, not both zero, such that aK1(P)+ bK2(P) is a non-negative8

integer for every point P. Then K = aK1 +bK2 is a multiarc with parameters (n,w), where9

n = an1 +bn2 and w ≤ aw1 +bw2. The following special cases are of particular interest:10

• K = aK1 - a replicated arc.11

• K =−K1 +bχP , where b = maxP K1(P) (usually, K1 is considered as a minihyper –12

the minihyper associated with K ).13

A very important instance of the sum of multisets construction is the following. Let Si,14

i = 1, . . . ,h, be subspaces of PG(N,q) with dimSi = λi, then the multiset F = ∑hi=1 χSi is a15

minihyper with parameters16

(h

∑i=1

vλi+1,h

∑i=1

vλi ;N,q).17

If s = maxP∈P (∑hi=1 χSi(P)), then the multiset K = sχP −F is a multiarc with parameters18

(svN+1−h

∑i=1

vλi+1,svN −h

∑i=1

vλi ;N,q).19

We will discuss this construction in more detail in Section 4 when we describe the Belov-20

Logachev-Sandimirov construction for linear codes meeting the Griesmer bound.21

Restriction to a subspace22

Let K : P → N be an (n,w)-multiarc in PG(N,q) and let U be an u-dimensional subspace23

of PG(N,q). The restriction of K to U is defined by24

K |U : P (U)→ N : P 7→ K (P).25

Then K |U is an (n′,w′)-multiarc in PG(u,q), with n′ = K (U). For the value of w′, we can26

give only a bound.27

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Projections of arcs1

Let K be an (n,w;N,q)-multiarc. Fix an u-dimensional subspace U in PG(N,q). Let2

furthermore V be a v-dimensional subspace in PG(N,q), with u+v = N−1 and U ∩V = /0.3

Define the projection ϕ = ϕU,V from U onto V by4

ϕU,V : P \U →V : P 7→V ∩〈U,P〉,5

where P is the point set of PG(N,q). Note that ϕU,V maps (u + s)-dimensional subspaces6

containing U into (s−1)-dimensional subspaces contained in V . The induced multiarc K ϕ7

is defined on the points of V by8

K ϕ : P (V )→ N : P 7→ ∑Q∈P\U : ϕU,V (Q)=P

K (Q).9

If S is a t ′-dimensional subspace in V , then K ϕ(S) = K (〈S,U〉)−K (U). Here, 〈S,U〉10

denotes the projective subspace of PG(N,q) generated by S and U . Clearly, K ϕ is an11

(n−K (U),w′−K (U))-multiarc in V ∼= PG(v,q), with w′ ≤w. Similarly, if K is an (n,w)-12

minihyper, then K ϕ is an (n−K (U),w′−K (U))-minihyper in V , with w′ ≥ w.13

The dual construction for arcs14

This construction is a generalization of familiar geometrical constructions to the case where15

multiple points are allowed. It has been introduced by Brouwer and van Eupen [9] for16

linear codes and formulated for multiarcs by Dodunekov and Simonis [25]. Let K be an17

(n,w;N,q)-multiarc and set W = {K (H) | H ∈ H }, where H is the set of all hyperplanes18

in PG(N,q). Let σ : W → N be a fixed mapping. The multiarc19

K σ : H → N : H 7→ σ(K (H))20

is called the σ-dual multiarc to K . Let (ai)i≥0 be the spectrum of K . Then the parameters21

of K σ are (n′,w′), where22

n′ = ∑i∈W

σ(i)ai, w′ = maxP

K σ(P) = maxP

∑H:P∈H,H∈H

K σ(H).23

Let σ(x) = αx + β, α,β ∈ Q, be a linear function, which takes on non-negative integer24

values for each x ∈W . Then25

K σ = K (α,β) = αK +βχ|H .26

〈thm:duality〉Theorem 2.3. Let K be an (n,w)-multiarc in PG(N,q). Then K (α,β) has parameters

n′ = αnqN −1q−1

+βqN+1−1

q−1,

w′ = maxP

{α

(n

qN−1−1q−1

+qN−1K (P))

+βqN −1q−1

},

where the maximum is taken over all points P in PG(N,q).27

Theorem 2.3 has been used repeatedly in the construction of various optimal arcs and28

codes [43, 55].29

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3 Arcs and linear MDS codes1

3.1 Introduction to arcs and linear MDS codes2

We now present the most famous example of the links between coding theory and Galois3

geometries, i.e., the link between linear MDS codes and arcs in Galois geometries. The4

chapter on linear MDS codes is described in [57] as one of the most fascinating in all of5

coding theory, and this is motivated by the many nice results on linear MDS codes obtained6

via the geometrical links with the arcs in Galois geometries.?? We first present the linear7

MDS codes, and then the arcs in Galois geometries.8

〈thm:Singleton〉Theorem 3.1 (The Singleton bound). For a linear [n,k,d]q code C, d ≤ n− k +1.9

Definition 3.2. A linear [n,k,d = n− k + 1]q code is called a linear Maximum Distance10

Separable (MDS) code.11

The following theorem gives the fundamental properties of linear MDS codes, which12

will enable us to make the links to the geometrically equivalent arcs in Galois geometries.13

〈thm:1:1〉Theorem 3.3. Let C be a linear [n,k,d]q code, then the following properties are equivalent:14

1. C is a linear [n,k,n− k +1]q MDS code,15

2. every k columns of a generator matrix G of C are linearly independent,16

3. every n− k columns of a parity check matrix H of C are linearly independent,17

4. C⊥ is a linear [n,n− k,k +1]q MDS code.18

Independently, the following concept of arcs was defined in Galois geometries [45].19

〈def:1:1〉Definition 3.4. An n-arc in PG(k− 1,q) is a set of n points, every k of which are linearly20

independent. An n-arc in PG(k−1,q) is called complete if and only if it is not contained in21

an (n+1)-arc of PG(k−1,q).22

Definition 3.4 immediately makes the link with Theorem 3.3 (2), which gives the fol-23

lowing equivalence.24

Theorem 3.5. The set K = {g1, . . . ,gn} is an n-arc in PG(k−1,q) if and only if the (k×n)25

matrix G = (g1 · · ·gn) defines a linear [n,k,n− k +1]q MDS code C.26

The equivalence of Theorem 3.3 (1) and Theorem 3.3 (4) now leads to the following27

geometrical result.28

Theorem 3.6. Let K = {g1, . . . ,gn} be an n-arc in PG(k−1,q) defining the linear [n,k,n−29

k + 1]q MDS code with generator matrix G = (g1 · · ·gn), then there exists an n-arc K =30

{h1, . . . ,hn} in PG(n− k−1,q) such that K defines the dual [n,n− k,k +1]q MDS code C⊥31

via the parity check matrix H = (h1 · · ·hn) of C.32

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So the existence of an n-arc K in PG(k− 1,q) implies the existence of an n-arc K in1

PG(n− k−1,q). We say that an n-arc K in PG(k−1,q) and an n-arc K in PG(n− k−1,q)2

are C-dual if and only if they define dual linear MDS codes.3

The standard example of an n-arc in PG(k−1,q) is the normal rational curve.4

Definition 3.7. A normal rational curve K in PG(k− 1,q), 2 ≤ k ≤ q− 1, is a (q + 1)-5

arc projectively equivalent to the set of points {(1, t, . . . , tk−1)|t ∈ F+q };F+

q = Fq∪{∞},∞ /∈6

Fq, t = ∞ defines the point (0, . . . ,0,1).7

The normal rational curves define the classical examples of linear MDS codes, i.e., the8

Generalized Doubly-Extended Reed-Solomon (GDRS) codes. These GDRS codes are used9

to encode music on compact disc.10

We also wish to mention that a particular non-GDRS [8,4,5]256 code is used in the11

Advanced Encryption Standard (see e.g. [58]).12

3.2 The largest arcs in Galois geometries13

The maximum number of points in an n-arc of PG(k−1,q) is denoted by m(k−1,q). The14

problem of determining the exact value of m(k−1,q) and of characterizing the m(k−1,q)-15

arcs in PG(k− 1,q) has been in the focus of research on arcs and linear MDS codes. We16

now state the main results on this central point of research.17

3.3 Arcs in PG(2,q)18

q m(2,q)q even q+2 [7]q odd q+1 [7]

Table 1: m(2,q)

An m(2,q)-arc in PG(2,q), q odd, is called an oval, and an m(2,q)-arc in PG(2,q), q19

even, is called a hyperoval. The following theorem of B.Segre inspired and motivated many20

researchers to investigate substructures in Galois geometries.21

Theorem 3.8. (Segre [64]) For q odd, an oval is the set of rational points of a conic.?〈thm:1:2〉?

22

The classical example of a hyperoval in PG(2,q), q even, is a conic plus its nucleus (the23

intersection point of its tangents). A hyperoval of this type is called regular. As shown by24

Segre [65], for q = 2,4,8, every hyperoval is regular.25

For q = 2h,h ≥ 4, there exist irregular hyperovals, that is, hyperovals which are not26

the union of a conic and its nucleus. Several infinite classes of irregular hyperovals are27

known. The problem of classifying the hyperovals in PG(2,q), q even, is one of the hardest28

problems in Galois geometries. In general, the following result is valid.29

Theorem 3.9. (Segre [65]) Any hyperoval of PG(2,q), q = 2h and h > 1, is projectively30

equivalent to a hyperoval31

D(F) = {(1, t,F(t))|t ∈ Fq}∪{(0,1,0),(0,0,1)},32

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where F is a permutation polynomial over Fq of degree at most q− 2, satisfying F(0) =1

0, F(1) = 1, and such that Fs(X) = (F(X + s)+F(s))/X is a permutation polynomial for2

each s in Fq, satisfying Fs(0) = 0.3

We refer to [49, Table 2.2] for the known infinite classes of hyperovals. Particular ex-4

amples include the translation hyperovals D(F) = {(1, t, t2i)|t ∈ Fq}∪{(0,1,0),(0,0,1)},5

with q = 2h, gcd(i,h) = 1.6

In the next tables, we state results on the m(k−1,q)-arcs in PG(k−1,q), for k ≥ 3. In7

many cases, we have chosen to state only one bound for q odd and for q even, so that we8

can explain the results in more detail. For the best known bounds, we refer to the tables9

of [49]. A large number of these results rely on the problem of finding the size m′(2,q) of10

the second largest complete arcs in PG(2,q). We first mention some of the known results,11

and then give a brief description on how these results were obtained.12

In Table 2, for q subject to the conditions in the first column, the second column gives13

an upper bound on m′(2,q), and the third column indicates when this upper bound is sharp.14

The fourth column gives the value of m(2,q). So any n-arc, with n > m′(2,q), is contained15

in an m(2,q)-arc. The last column describes the type of the m(2,q)-arc.16

q m′(2,q) Sharp m(2,q) m(2,q)-arcq = ph, p ≥ 5 ≤ q−√q/2+5 q+1 conic [46]q = 22e, e > 1 = q−√q+1 yes q+2 hyperoval [6, 28, 50, 66]

q = 22e+1, e ≥ 1 ≤ q−√

2q+2 q = 8 q+2 hyperoval [72]

Table 2: Upper bounds on m′(2,q)〈tab:ubmd〉

〈tech:1〉Techniques 3.10. The results of Table 2 were found by using the following method. Con-17

sider an n-arc K = {`1, . . . , `n} of lines in PG(2,q), i.e., a set of n lines, no three concurrent.18

Then every line `i contains q+2−n points Pi1, . . . ,Pi,q+2−n not lying on a second line of K.19

It has been proven that all these n(q + 2− n) points Pi j, i = 1, . . . ,n, j = 1, . . . ,q + 2− n,20

belong to an algebraic curve Γ of degree q+2−n when q is even, and of degree 2(q+2−n)21

when q is odd [45]. By using bounds on the number of points and other fundamental prop-22

erties of these algebraic curves, it was shown that Γ contains linear components over Fq;23

linear components which extend the given n-arc K to a larger arc. So, here in this context,24

algebraic geometry plays a fundamental role.25

Based on the sharpness of the second bound in Table 2, the following conjecture has26

been stated.27

Conjecture 3.11. m′(2,q) = q−√q+1 for q = p2e, q > 9.28

For arcs in PG(2,q), q = 22e, q > 4, of size smaller than q−√q + 1, there is the29

following result of Hirschfeld and Korchmáros [47].30

Theorem 3.12. A complete n-arc of PG(2,q), q = 22e, e > 2, has size q + 2, q−√q + 1,31

or at most size q−2√

q+6.32

ii


ii

ii

Uncor

rected

Proof


3.4 Results in higher dimensions1

In Tables 3 and 4, NRC stands for normal rational curve. In PG(3,q), q = 2h, h > 2,2

Le ={(1, t, te, te+1)|t ∈ F+

q}

, with e = 2v, gcd(v,h) = 1, and with t = ∞ defining the point3

e3 = (0,0,0,1). Let e0 = (1,0, . . . ,0),e1 = (0,1,0, . . . ,0), . . . ,eN = (0, . . . ,0,1), and let4

e = (1, . . . ,1).5

Table 3 shows the value of m(N,q) for small dimensions N. The characterization of the6

m(N,q)-arcs L in PG(N,q) is given in the fourth column.7

q N m(N,q) m(N,q)-arc Lq odd , q > 3 3 q+1 NRC [63]q = 2h, q > 4 3 q+1 Le, e = 2v, gcd(v,h) = 1 [14, 15, 33]q odd, q > 5 4 q+1 NRC for q > 83 [63]q = 2h, q > 4 4 q+1 NRC [14, 16, 33]

q N ≥ q−1 N +2 {e0, . . . ,eN ,e} [13]

Table 3: m(N,q) and m(N,q)-arcs〈tab:mnq〉

Table 4 summarizes the main results on the extendability of n-arcs in PG(N,q) to larger8

arcs. An n-arc in PG(N,q), satisfying the condition on n in the third column, can be ex-9

tended uniquely to a (q+1)-arc L, whose description is given in column 4. The results are10

respectively due to Hirschfeld and Korchmáros for the first formula [46], and to Storme and11

Thas [69] for the last two formulas.12

N q n > L≥ 2 q = ph, p ≥ 5 q−√q/2+N +3 NRC3 q = 2h,h > 1 q−√q/2+9/4 Le

≥ 4 q = 2h, h > 2 q−√q/2+N−3/4 NRC

Table 4: Upper bounds on m′(N,q)〈tab:ubmdn〉

Techniques 3.13. The principal arguments for obtaining these extension results are as13

follows.14

For q odd, projection arguments can be used. Consider an n-arc K = {P1, . . . ,Pn} in15

PG(N,q). Project K \{Pn} from the point Pn onto a hyperplane Π not passing through Pn,16

then an (n− 1)-arc K′ in Π is obtained. By selecting the lower bound on n in the correct17

way, as was done in Table 4, by induction on N, it is known that this (n− 1)-arc in Π is18

contained in a normal rational curve of Π; hence, K belongs to a cone with vertex Pn and19

base a normal rational curve in a hyperplane Π of PG(N,q). But Pn is an arbitrary point20

of K, so in fact, K is contained in the intersection of n such cones. This implies that K itself21

is contained in a normal rational curve of PG(N,q).22

For n-arcs in PG(3,q), q even, the arguments for q odd cannot be used, since there is23

no classification of hyperovals in PG(2,q), q even. So a completely different technique had24

to be developed [12]. Here, consider an n-arc K of planes in PG(3,q), then it is possible25

to associate an algebraic surface Φ of degree q + 3− n to K. For large n, it was shown26

ii


ii

ii

Uncorr

ected

Proof


that this surface Φ contains at least one plane Π, and this plane Π extends K to a larger1

(n+1)-arc.2

Similarly, to an n-arc K in PG(4,q), q even, an algebraic hypersurface Φ of degree3

q + 4− n can be associated. For large n, it is proven that Φ contains a hyperplane Π;4

where Π extends K to an (n+1)-arc.5

Since it was proven that every (q+1)-arc in PG(4,q), q even, q≥ 8, is a normal rational6

curve (Table 3), for the characterization of large arcs in PG(N,q), q even, N > 4, the7

projection arguments used for q odd can now be applied here.8

Techniques 3.14. As indicated in the beginning of this section, an n-arc in PG(k− 1,q)9

defines a C-dual n-arc in PG(n−k−1,q). This C-duality principle of arcs makes it possible10

to translate results on n-arcs to results on C-dual n-arcs. The results in the preceding tables11

on arcs in PG(N,q), with N small, immediately imply other results on arcs in PG(N,q),12

where N is close to q. We now present a number of these results.13

In Table 5, in the spaces PG(N,q), with N satisfying the bounds in the table, any n-arc,14

where n satisfies the bound in the second column, is contained in a normal rational curve.15

So, in these spaces, m(N,q) = q+1 and every (q+1)-arc is a normal rational curve.16

q n ≥ m(N,q) = q+1 and (q+1)-arc = NRCq = p2e, p > 2, e ≥ 1 N +4 q−3 ≥ N > q−√q/4−39/16

q = ph, p ≥ 5 N +4 q−3 ≥ N > q−√q/2+1q = 2h, h > 2 N +6 q−5 ≥ N > q−√q/2−11/4

Table 5: Arcs in PG(N,q), N close to q〈tab:arcsnq〉

Up to now, all presented results for q odd state that a (q+1)-arc in PG(k−1,q), q odd,17

2 ≤ k ≤ q− 1, is a normal rational curve. So the conjecture arose that this is indeed the18

case. However, Glynn found a counterexample to this conjecture.19

Theorem 3.15. (Glynn [30]) In PG(4,9), a 10-arc is one of two types; it is either a nor-20

mal rational curve or it is equivalent to the 10-arc L = {(1, t, t2 + ηt6, t3, t4)|t ∈ F9} ∪21

{(0,0,0,0,1)}, where η4 =−1.22

We already have observed that there are fundamental differences between arcs in spaces23

of even characteristic and of odd characteristic; one of the differences consists of the (q+2)-24

arcs in PG(q−2,q), q even.25

Theorem 3.16. (Thas [71]) In PG(q−2,q), q even, m(q−2,q) = q+2.26

Theorem 3.17. (Storme and Thas [70]) In PG(q−2,q), q even, a point P = (a0, . . . ,aq−2)27

extends the normal rational curve K = {(1, t, . . . , tq−2)|t ∈ F+q } to a (q+2)-arc if and only28

if F(X) = ∑q−2i=0 aq−2−iX i+1 defines a (q + 2)-arc K′ = {(1, t,F(t))|t ∈ Fq} ∪ {e1,e2} in29

PG(2,q); in this case, K′ is a C-dual (q+2)-arc of K∪{P}.30

Table 6 presents the results of Storme and Thas [68] for the values of n for which there31

exist complete n-arcs in the respective spaces PG(N,q), q even, N large.32

ii


ii

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Uncorr

ected

Proof


q N n ∈q = 2h, q ≥ 32, q 6= 64 q−5 ≥ N > q−√q/2−11/4 {N +4,N +5,q+1}

q = 64 N = 58 or N = 59 {N +4,q+1}q = 2h, q ≥ 8 q−4 {q,q+1}q = 2h, q ≥ 8 q−3 {q+1}q = 2h q ≥ 4 q−2 {q+2}

Table 6: Spectrum of complete arcs〈tab:spectrum〉

3.5 Open problems1

1. The key tool in obtaining the results on the extendability of large n-arcs in PG(2,q) to2

ovals and hyperovals is the link between n-arcs K of lines in PG(2,q) and algebraic3

curves of degree 2(q+2−n) for q odd, and of degree q+2−n for q even (Techniques4

3.10). Continuing this study of algebraic curves is of great interest. See [48] for5

detailed information on algebraic curves over a finite field.6

2. The problem of the classification of the hyperovals in PG(2,q), q even, has been7

one of the earliest problems investigated in Galois geometries. The complete clas-8

sification of hyperovals has not yet been obtained. So we propose to investigate the9

classification problem of hyperovals in PG(2,q), q even.10

3. The problem of constructing large n-arcs in PG(2,q), different from ovals and hy-11

perovals, still merits attention. For q a square, we know the existence of complete12

(q−√q+1)-arcs [6, 28, 50, 66].13

Apart from this example, for general q, all the other known largest complete arcs14

have size at most approximately (q+1+2√

q)/2. The main constructions consist of15

half of the points of absolutely irreducible cubic curves, and of half of the points of a16

conic, to which some other points not lying on this conic are added. So we propose to17

investigate the construction of complete n-arcs in PG(2,q), with n > (q+1+2√

q)/2.18

4 Minihypers and the Griesmer bound19

〈section:Griesmer〉4.1 A geometrical proof of the Griesmer bound20

Let K be an (n,w;k− 1,q)-arc. We start with the observation that w points generate a21

subspace of projective dimension at most w−1, or, in other words, the maximal multiplicity22

of a subspace of dimension u is at least u+1. Hyperplanes have projective dimension k−2,23

therefore w ≥ k− 1. This is easily seen to be equivalent to the Singleton bound (Theorem24

3.1). The Griesmer bound is a generalization of the Singleton bound.25

Theorem 4.1. ( [32, 67]) For every linear [n,k,d]q code,?〈thm:griesmer〉?

26

n ≥k−1

∑i=0d d

qi e= gq(k,d).27

ii


ii

ii

Uncorr

ected

Proof


Proof. By induction on k. For the case k = 2, consider a multiarc K with parameters1

(n,w;1,q), w = n− d. On the projective line, fix a point P of maximal multiplicity w.2

There is a point Q on the projective line, Q 6= P, that has multiplicity K (Q)≥ d(n−w)/qe.3

Since P is of maximal multiplicity, we have w ≥ d n−wq e which implies n ≥ d + dd

qe.4

Assume the inequality has been proven for multiarcs in the projective geometries5

PG(N,q), N ≤ k− 2. Consider an (n,w;k− 1,q)-multiarc K . There exists a hyperplane6

H of multiplicity n− d. The sum of the multiplicities of the points outside of H is d.7

Since the number of points outside of H is qk−1, there exists a point P (P 6∈ H) such that8

K (P) ≥ dd/qk−1e. Consider a projection ϕ from P onto some hyperplane not incident9

with P. The induced multiarc K ϕ has parameters (n′,w′;k− 2,q), where n′ = n−K (P)10

and w′ ≤ w−K (P). Note that n′−w′ ≥ n−w = d. Hence, by the induction hypothesis,11

n−K (P) = n′ ≥ ∑k−2i=0 d

dqi e.12

Linear codes attaining the Griesmer bound, i.e. with parameters [gq(k,d),k,d]q, are13

called Griesmer codes.14

4.2 Minihypers and the Belov-Logachev-Sandimirov construction15

The link between minihypers in PG(k−1,q) and linear [n,k,d]q codes meeting the Griesmer16

bound is described in the following way.17

For (s− 1)qk−1 < d ≤ sqk−1, d can be written uniquely as d = sqk−1 −∑hi=1 qλi such18

that:19

(a) 0 ≤ λ1 ≤ ·· · ≤ λh < k−1,20

(b) at most q−1 of the values λi are equal to a given value.21

Using this expression for d, the Griesmer bound for a linear [n,k,d]q code can be ex-22

pressed as:23

n ≥ svk−h

∑i=1

vλi+1.24

Hamada and Helleseth showed that in the case d = sqk−1−∑hi=1 qλi , there is a one-to-25

one correspondence between the set of all non-equivalent [n,k,d]q codes meeting the Gries-26

mer bound and the set of all projectively distinct (∑hi=1 vλi+1, ∑

hi=1 vλi ;k−1,q)-minihypers27

F [37].28

Belov, Logachev, and Sandimirov [3] gave a construction method for Griesmer codes,29

which is easily described by using the corresponding minihypers.30

Consider in PG(k− 1,q) a sum of ε0 points P1,P2, . . . ,Pε0 , ε1 lines `1, `2, . . ., `ε1 , . . . ,31

εk−2 (k−2)-dimensional subspaces π(k−2)1 , . . . ,π

(k−2)εk−2 , with 0 ≤ εi ≤ q−1, i = 0, . . . ,k−2,32

then such a sum defines a (∑k−2i=0 εivi+1,∑

k−2i=0 εivi;k− 1,q)-minihyper F , where the multi-33

plicity of a point R of PG(k−1,q) equals the number of objects, in the description above,34

in which it is contained (See also the sum of multisets in Subsection 2.5.).35

Now that the standard examples of minihypers are known, the characterization problem36

on minihypers, and equivalently on linear codes meeting the Griesmer bound, arises:37

ii


ii

ii

Uncor

rected

Proof


Characterize ( f ,m;k−1,q)-minihypers F for given parameters f = ∑k−2i=0 εivi+1, m =1

∑k−2i=0 εivi,k, and q.2

Fundamental research on this problem was performed by Hamada et al. who, in many3

articles, obtained a lot of results on minihypers and who developed a great amount of tech-4

niques useful in the study of minihypers. Their main results are in [36, 38].5

Improvements to the results of [36, 38] were found by for instance De Beule, Metsch,6

and Storme.7

Theorem 4.2. (De Beule, Metsch, and Storme [22]) A projective (∑k−2i=0 εivi+1,∑

k−2i=0 εivi;k−

〈JDBKMLS_minihypers〉8

1,q)-minihyper, where ∑k−2i=0 εi ≤ δ0 with δ0 equal to one of the values in Table 4.1, is a union9

of εk−2 hyperplanes, εk−3 (k−3)-dimensional spaces, . . . ,ε1 lines, and ε0 points, which all10

are pairwise disjoint, so is of Belov-Logachev-Sandimirov type.11

In the following table, q = ps, p prime, s ≥ 1.12

p s δ0

p 1 ≤ (p+1)/2p 3 ≤ p2

p even ≤√q

2 6m+1,m ≥ 1 ≤ 24m+1−24m−22m+1/2> 2 6m+1,m ≥ 1 ≤ p4m+1− p4m− p2m+1/2+1/22 6m+3,m ≥ 1 < 24m+5/2−24m+1−22m+1 +1

> 2 6m+3,m ≥ 1 ≤ p4m+2− p2m+2 +2≥ 5 6m+5,m ≥ 0 < p4m+7/2− p4m+3− p2m+2/2+1

Table 7: Upper bounds on δ0

Regarding characterization results on weighted minihypers, we mention the following13

two results. The next theorem was first proven by Hamada for projective minihypers, while14

the second theorem is the weighted version of a result of Hamada, Helleseth, and Maekawa15

[36, 38].16

Theorem 4.3. (Hamada [34, 35] and Landjev and Storme [54]) A (∑hi=1 vλi+1,∑

hi=1 vλi ;k−17

1,q)-minihyper, with k−1 > λ1 > λ2 > · · ·> λh ≥ 0, is the sum of a λ1-dimensional space,18

a λ2-dimensional space, . . ., and a λh-dimensional space.19

?〈Maekawa_weighted〉?Theorem 4.4. (De Beule, Metsch, and Storme [23]) A (∑k−2i=0 εivi+1, ∑

k−2i=0 εivi; k− 1,q)-20

minihyper, where ∑k−2i=0 εi <

√q+1, is a sum of εk−2 hyperplanes, εk−3 (k−3)-dimensional21

spaces, . . . ,ε1 lines, and ε0 points, so it is of Belov-Logachev-Sandimirov type.22

Techniques 4.5. The results on the minihypers are obtained via a variety of techniques.23

First of all, minihypers are particular examples of blocking sets (see e.g. [5]). Hence,24

characterization results on minimal blocking sets play a crucial role in the charac-25

terization of minihypers. The characterization of a minihyper is in many cases ob-26

tained by building up the minihyper in inductive steps. As particular example, a Belov-27

Logachev-Sandimirov (∑k−2i=0 εivi+1, ∑

k−2i=0 εivi; k− 1,q)-minihyper, ∑

k−2i=0 εi small, which is28

the union of εk−2 hyperplanes, εk−3 (k − 3)-dimensional spaces, . . . ,ε1 lines, and ε029

ii


ii

ii

Uncor

rected

Proof


points, which are pairwise disjoint, can be characterized in the following way. First1

of all, (ε1(q + 1) + ε0,ε1;k − 1,q)-minihypers, ε1 + ε0 small, are characterized as the2

union of ε1 lines and ε0 points. Once this is done, this result is used to characterize3

(ε2(q2 + q + 1) + ε1(q + 1) + ε0,ε2(q + 1) + ε1;k− 1,q)-minihypers, ε2 + ε1 + ε0 small,4

as the union of ε2 planes, ε1 lines, and ε0 points, which are pairwise disjoint. Namely,5

many hyperplane intersections of (ε2(q2 +q+1)+ ε1(q+1)+ ε0,ε2(q+1)+ ε1;k−1,q)-6

minihypers are (ε′1(q + 1) + ε′0,ε′1;k− 2,q)-minihypers, which are already characterized.7

So, these hyperplane intersections are exactly known. This then is used to characterize the8

(ε2(q2 + q + 1)+ ε1(q + 1)+ ε0,ε2(q + 1)+ ε1;k−1,q)-minihypers, ε2 + ε1 + ε0 small, as9

the union of ε2 planes, ε1 lines, and ε0 points. Once this is done, inductive arguments can10

be used to characterize (∑k−2i=0 εivi+1, ∑

k−2i=0 εivi; k− 1,q)-minihypers, ∑

k−2i=0 εi small, as the11

union of εk−2 hyperplanes, εk−3 (k− 3)-dimensional spaces, . . . ,ε1 lines, and ε0 points,12

which are pairwise disjoint.13

Polynomial techniques play a central role in the study of blocking sets, see e.g. [5]14

and [2]. So, it is worth considering the polynomial techniques for the study of minihypers.15

In particular, in obtaining the results of Theorem 4.2, also polynomial techniques were used.16

Recently, characterizations of minihypers involving Baer subgeometries in PG(N,q), q17

square, have been obtained. For particular results, we refer to [31].18

5 Saturating sets in Galois geometries and covering radius19

Definition 5.1. Let C be a linear [n,k,d]q code. The covering radius of the code C is the20

smallest integer R such that every n-tuple in Fnq lies at Hamming distance at most R from a21

codeword in C.22

The following theorem will be the basis for making the link with the geometrically23

equivalent objects of the saturating sets in Galois geometries.24

〈th:3〉Theorem 5.2. Let C be a linear [n,k,d]q code with parity check matrix H = (h1 · · ·hn).25

Then the covering radius of C is equal to R if and only if every (n− k)-tuple over Fq26

can be written as a linear combination of at most R columns of H.27

Definition 5.3. Let S be a subset of PG(N,q). The set S is called ρ-saturating when every28

point P from PG(N,q) can be written as a linear combination of at most ρ+1 points of S.29

Taking into account Theorem 5.2, the preceding definition means that: ρ-saturating30

sets S in PG(n− k−1,q) determine the parity check matrices of linear [n,k,d]q codes with31

covering radius R = ρ+1.32

Example 5.4. The linear codes with covering radius R = 2 and with minimum distance33

d ≥ 4 are important examples. Such a code has a parity check matrix whose columns define34

an n-cap K = {h1, . . . ,hn} in PG(n− k− 1,q) (see e.g. [26]). The fact that the covering35

radius R is equal to two signifies that every point from PG(n− k−1,q)\K can be written36

as a linear combination of two columns of H, i.e., that every point of PG(n− k− 1,q) \K37

is linearly dependent on two columns of H. This signifies also that no point of PG(n− k−38

1,q)\K extends the n-cap K in PG(n− k−1,q) to an (n + 1)-cap. Hence, this altogether39

ii


ii

ii

Uncor

rected

Proof


proves that the columns of a parity check matrix H of a linear [n,k,d]q code, with d ≥ 41

and R = 2, define a complete n-cap of PG(n−k−1,q). In this way, the complete caps K in2

a projective space PG(N,q) are particular examples of 1-saturating sets.3

In the study of ρ-saturating sets, one of the most important research problems is the4

problem of finding ρ-saturating sets of the smallest possible cardinality. The cardinality of5

a smallest possible set S from PG(N,q) which is ρ-saturating is denoted by the parameter6

k(N,q,ρ).7

We now present a number of the known upper bounds on the parameter k(N,q,ρ).8

Regarding the parameters k(N,q,1), good upper bounds on k(N,q,1) have been found9

by the construction of small complete caps. In the following table, the first two results are10

of Davydov and Tombak [29], the next two results of Pambianco and Storme [62], and the11

last two results of Davydov and Östergard [21]. In the upper bounds of the following table,12

the parameter n2(N,q) denotes the smallest cardinality of a complete cap in PG(N,q).13

N q k(N,q,1),n2(N,q)2k 2 ≤ 23 ·2k−3−3

2k +1 2 ≤ 30 ·2k−3−32k q = 2h ≥ 4 ≤ qk +3(qk−1 +qk−2 + · · ·+q)+2

2k +1 q = 2h ≥ 4 ≤ 3(qk +qk−1 + · · ·+q)+24k +2 q ≥ 5 odd ≤ q2k+1 +n2(2k,q)4k +2 q ≥ 9 odd q2k+1− (q+1)+n2(2k,q)+n2(2,q)

Table 8: Upper bounds on k(N,q,1) n2(N,q)

Other upper bounds on k(N,q,ρ) were given by Davydov and Östergard [17–20]. A14

number of these upper bounds are mentioned in the next theorem.15

〈th:5〉Theorem 5.5. (1) For p ≥ 2 and m ≥ 2, k(2, pm,1)≤ 2pm−1 +2.16

(2) For q ≥ 4, k(3,q,1)≤ 2q+1.17

(3) For p ≥ 2 and m ≥ ρ+1, k(ρ+1, pm,ρ)≤ (p−1)(

ρ+12

)+ pm−ρ(ρ+1)+1.18

(4) For q 6= 3, k(5,q,2)≤ 3q+1.19

Techniques 5.6. (1) Of particular interest to these results is the fact that these upper bounds20

were obtained by the construction of carefully selected subsets of points from PG(N,q). The21

1-saturating sets and 2-saturating sets of Theorem 5.5 (2) and Theorem 5.5 (4) are defined22

by the columns of the following two matrices H1 and H2:23

H1 =

1 · · · 1 0 0 0 · · · 0a1 · · · aq 1 0 0 · · · 0a2

1 · · · a2q 0 0 1 · · · 1

0 · · · 0 0 1 a2 · · · aq

24

ii


ii

ii

Uncor

rected

Proof


and1

H2 =

1 · · · 1 0 0 · · · 0 0 · · · 0 0a1 · · · aq 1 0 · · · 0 0 · · · 0 0a2

1 · · · a2q 0 1 · · · 1 0 · · · 0 0

0 · · · 0 0 a2 · · · aq a21 · · · a2

q 00 · · · 0 0 0 · · · 0 a1 · · · aq 10 · · · 0 0 0 · · · 0 1 · · · 1 0

,2

with Fq = {a1 = 0,a2, . . . ,aq}.3

These two particular examples show that by taking the unions of particularly selected4

subsets of Galois geometries, such as lines and conics, it is possible to obtain very good5

upper bounds on the parameter k(N,q,ρ). In the matrix H1, the first q columns are points6

of a conic and the last q columns are points of a line. In the matrix H2, we recognize q7

points of two conics, and q−1 points of a line.8

(2) In [17, 18, 20], Davydov and Östergard also show how, by means of inductive con-9

structions, it is possible to construct infinite classes of ρ-saturating sets.10

5.1 Open problems11

1. In the articles of Davydov and Östergard, a lot of attention has been paid to 2-12

saturating and 3-saturating sets. It is of great interest to construct small ρ-saturating13

sets, with ρ > 3.14

2. Which particular subsets of Galois geometries, or unions of carefully selected subsets15

of Galois geometries, define small ρ-saturating sets?16

3. Which inductive construction methods lead to interesting infinite classes of small17

ρ-saturating sets?18

6 Extension results19

6.1 The extension result of Hill and Lizak20

We start this section by formulating two theorems on blocking sets that have become clas-21

sical. Interestingly, these results are related to the extendability problem for linear codes.22

The first theorem is of Bose and Burton [8].23

Theorem 6.1. (Bose and Burton [8]) Let K be an (n,1)-blocking set in PG(N,q) with〈thm:bose-burton〉

24

respect to the s-dimensional subspaces that has the smallest possible cardinality. Then25

n = vN−s+1 and K = χF , where F is a fixed (N− s)-dimensional subspace of PG(N,q).26

An (n,1)-blocking set with respect to s-dimensional subspaces in PG(N,q) is called27

non-trivial if there exists no (N − s)-dimensional subspace δ with K (P) > 0 for every28

point P on δ. The next result which was proven independently by Beutelspacher [4] and29

Heim [39] characterizes the smallest non-trivial blocking sets.30

ii


ii

ii

Uncor

rected

Proof


Theorem 6.2. (Beutelspacher and Heim [4, 39]) The smallest non-trivial (n,1)-blocking〈thm:nontriv〉

1

sets in PG(N,q) with respect to the s-dimensional subspaces are cones with an (N− s−2)-2

dimensional vertex and a non-trivial (n′,1;2,q)-blocking set of minimum cardinality in a3

plane of PG(N,q) as base curve. Consequently,4

n = qN−s +qN−s−1 + · · ·+1+qN−s−1 · r(q),5

where q+ r(q)+1 is the minimal size of a non-trivial blocking set in PG(2,q).6

Now we turn to the extendability problem for linear codes and arcs. It has been long7

known that a binary [n,k,d] code of odd minimum distance can be extended to an [n +8

1,k,d + 1] code by adding a parity check. This result has been generalized by Hill and9

Lizak in [41, 42].10

〈thm:HL〉Theorem 6.3. (Hill and Lizak [41, 42]) Let C be an [n,k,d]q code with gcd(d,q) = 1 and11

with all non-zero weights congruent to 0 or d (mod q). Then C can be extended to an12

[n+1,k,d +1]q code.13

The geometrical version of this result is given below. We include a proof which relies14

on the Bose-Burton theorem (Theorem 6.1).15

〈thm:HLg〉Theorem 6.4. Let K be an (n,w;k− 1,q)-arc with gcd(n−w,q) = 1. Assume that the16

multiplicities of all hyperplanes are congruent to n or w (mod q). Then K can be extended17

to an (n+1,w;k−1,q)-arc.18

Proof. Fix a hyperplane H0 in PG(k−1,q) with K (H0) = w. For any subspace δ of codi-19

mension 2, δ ⊂ H0, consider the hyperplanes Hi, i = 0, . . . ,q, containing δ. Let α of them20

be of multiplicity congruent to n (mod q). Then21

n =q

∑i=0

K (Hi)−qK (δ)≡ αn+(q+1−α)w (mod q),22

whence (α− 1)(n−w) ≡ 0 (mod q) and α = 1. Hence, the number of hyperplanes of23

multiplicity congruent to n (mod q) equals the number of subspaces of codimension 1 in H024

and forms a blocking set in the dual space. By Theorem 6.1, a blocking set in PG(k−1,q)25

with respect to the lines having cardinality (qk−1 − 1)/(q− 1) consists of the points of a26

hyperplane. By duality, this implies that all hyperplanes of multiplicity congruent to n27

(mod q) pass through a fixed point P. Moreover, these are all the hyperplanes through P.28

Hence, we can get an (n+1,w;k−1,q)-arc by increasing the multiplicity of P by 1.29

Using the result of Beutelspacher and Heim (Theorem 6.2), we can go a bit further.30

Theorem 6.5. (Landjev and Rousseva [53]) Let K be an (n,w;k−1,q)-arc, q = ps, with〈thm:ext〉

31

spectrum (ai)i≥0. Let w 6≡ n (mod q) and32

∑i6≡w (mod q)

ai < qk−2 +qk−3 + · · ·+1+qk−3 · r(q), (1)33

where q + r(q)+ 1 is the minimal size of a non-trivial blocking set of PG(2,q). Then K is34

extendable to an (n+1,w;k−1,q)-arc.35

ii


ii

ii

Uncorr

ected

Proof


From this theorem, we can derive a useful corollary which roughly says that if for a1

given (n,w;k− 1,q)-arc with w ≡ n + 1 (mod q), there are not too many hyperplanes of2

multiplicity 6≡ n,n+1 (mod q), then this arc is extendable.3

Theorem 6.6. (Landjev and Rousseva [53]) Let K be a non-extendable (n,w;k−1,q)-arc,?〈thm:ext1〉?

4

k≥ 3, q = ps, with gcd(n−w,q) = 1 and with spectrum (ai)i≥0. Let H be a hyperplane with5

K (H) ≡ w (mod q) and denote by θ the maximal number of hyperplanes of multiplicity6

6≡ w (mod q) that are incident with a subspace of codimension 2 contained in H. Then7

∑i6≡n,w (mod q)

ai ≥ qk−3 · r(q)/(θ−1),8

where r(q) is the same as in Theorem 6.5.9

This result is easily restated for linear codes.10

?〈thm:ext2〉?Theorem 6.7. Let C be a non-extendable [n,k,d]q code, q = ps, with gcd(d,q) = 1. If11

(Ai)i≥0 is the spectrum of C, then ∑i6≡0,d (mod q) Ai ≥ qk−3 · r(q), where r(q) is the same as12

in Theorem 6.5.13

6.2 Diversity and extendability14

In a series of papers, Maruta further generalized these results [59–61]. Let C be an [n,k,d]q15

code with k ≥ 3 and with gcd(d,q) = 1, and with spectrum (Ai)i≥0. We define16

Φ0 =1

q−1 ∑q|i,i6=0

Ai, Φ1 =1

q−1 ∑i6≡0,d (mod q)

Ai.17

The pair (Φ0,Φ1) is called the diversity of C. The theorem of Hill and Lizak (Theorem 6.3)18

states that every linear code with Φ1 = 0 is extendable.19

?〈thm:maruta_main〉?Theorem 6.8. Let q ≥ 5 be an odd prime power and let k ≥ 3 be an integer. For a linear20

[n,k,d]q code C with d ≡−2 (mod q) and with diversity (Φ0,Φ1) such that Ai = 0 for all21

i 6≡ 0,−1,−2 (mod q), the following results are equivalent:22

1. C is extendable.23

2. (Φ0,Φ1) ∈ {(vk−1,0),(vk−1,2qk−2),(vk−1 + (ρ − 2)qk−2,2qk−2)} ∪ {(vk−1 +24

iqk−2,(q−2i)2k−2 | i = 1, . . . ,ρ−1}, where ρ = (q+1)/2.25

Furthermore, if 1. and 2. are valid and if (Φ0,Φ1) 6= (vk−1 +(ρ−2)qk−2,2qk−2), then26

C is doubly extendable.27

6.3 Extension results depending on divisibility and quasi-divisibility28

Let C be a linear [n,k,d]q code. Following Ward [73], we call the integer ∆ > 1 a divisor of29

C if it is a common factor of all weights of C. The code C is called divisible if it has a divisor30

∆ > 1. This definition can be given in a geometrical setting. Let K be an (n,w;k−1,q)-arc.31

The integer ∆ > 1 is said to be a divisor of K if K (H)≡ n (mod ∆) for every hyperplane32

ii


ii

ii

Uncorr

ected

Proof


H in PG(k− 1,q). The arc K is divisible if it has a divisor. Clearly, ∆ is a divisor of the1

code C if and only if it is a divisor of the associated arc K .2

Ward proved in [73] a theorem on the divisibility of linear codes meeting the Griesmer3

bound. Below we reformulate Ward’s result for Griesmer arcs, i.e. arcs associated with4

Griesmer codes.5

Theorem 6.9. (Ward [73]) Let K be a Griesmer (n,w)-arc in PG(k−1, p), p prime, with?〈thm:ward〉?

6

w ≡ n (mod pe), e ≥ 1. Then K (H)≡ n (mod pe) for every hyperplane H.7

For Griesmer arcs in projective geometries over non-prime fields (resp. Griesmer codes8

over such fields), we have the following weaker version of this result [73].9

Theorem 6.10. (Ward [73]) Let K be a Griesmer (n,w)-arc in PG(k−1,q), where q = pm,?〈thm:ward-nonprime〉?

10

p prime, m≥ 1, and let w≡ n (mod qe) for some integer e≥ 1. Then K (H)≡ n (mod pe)11

for every hyperplane H of PG(k−1,q).12

Let w,n be integers with w < n, gcd(w,n) = 1. The integer ∆ > 1 is called a quasi-divisor13

of the (n,w;k−1,q)-arc K if K (H)≡ n or w (mod ∆) for all hyperplanes in PG(k−1,q).14

An arc is called quasi-divisible if it has a quasi-divisor. The theorem of Hill-Lizak in its15

geometrical form (Theorem 6.4) says that if the (n,w;k− 1,q)-arc is quasi-divisible with16

∆ = q and with w≡ n+1 (mod q), then it is extendable to a divisible (n+1,w;k−1,q)-arc.17

The following theorem is useful if we know all the induced arcs of a given arc.18

Theorem 6.11. (Landjev and Rousseva [53]) Let K be an (n,w)-arc in PG(k−1,q), k≥ 4,?〈thm:quasi〉?

19

with n > w(q− 1). Denote by ϕP the projection from the point P. If the induced arc K ϕP20

has quasi-divisor q for every point P with K (P) > 0, then K is extendable.21

7 Codes arising from incidence matrices of Galois geometries22

7.1 Linear codes defined by incidence matrices of Galois geometries23

〈subs:incid1〉We define the incidence matrix A = (ai j) of points and hyperplanes in the projective space24

PG(N,q), q = ph, p prime, h≥ 1, as the matrix whose rows are indexed by the hyperplanes25

of PG(N,q) and whose columns are indexed by the points of PG(N,q), and with entry26

ai j ={

1 if point j belongs to hyperplane i,0 otherwise.

27

The p-ary linear code of points and hyperplanes of PG(N,q), q = ph, p prime, h≥ 1, is the28

Fp-span of the rows of the incidence matrix A. We denote this code by C(N,q). We identify29

the support of a codeword with the corresponding set of points of PG(N,q).30

The fundamental parameters n,k, and d of these linear codes C(N,q) are known [1,57]:31

1. n = qN +qN−1 + · · ·+q+1,32

2. k =(

p+N−1N

)h

+1,33

34

3. d = qN−1 + · · ·+q+1.35

ii


ii

ii

Uncor

rected

Proof


7.2 Small weight codewords1

We note that the minimum distance of this code C(N,q) is known. Moreover, the codewords2

of minimum weight also have been classified.3

Theorem 7.1. (Assmus and Key [1]) Every codeword of weight d = qN−1 + · · ·+ q + 1 in4

C(N,q) is, up to a scalar multiple, the incidence vector of a hyperplane.5

So the question arises: what is the second smallest weight of C(N,q), and what are the6

codewords of this second smallest weight: can they be characterized in a geometrical way?7

The difference of the incidence vectors of two hyperplanes of PG(N,q) defines a code-8

word of weight 2qN−1. So, the preceding questions can also be formulated in the following9

way: is 2qN−1 the second smallest weight of C(N,q); and if this is indeed the case, are10

all the codewords of weight 2qN−1 equal, up to a scalar multiple, to the difference of the11

incidence vectors of two hyperplanes?12

For the code C(2,q), it has effectively been proven that the second weight is equal to13

2q.14

Theorem 7.2. (Lavrauw et al. [56]) There are no codewords of weight in the interval [q+15

2,2q−1] in the linear code C(2,q).16

Techniques 7.3. To prove that 2q is the second smallest weight of the code C(2,q), q = ph,17

p prime, h ≥ 1, one can proceed in the following way.18

The minimum weight of C(2,q)∩C(2,q)⊥ is 2q [1]. So, only codewords in C(2,q) \19

C(2,q)⊥ having weight in the interval [q+2,2q−1] need to be eliminated.20

A possible codeword c ∈ C(2,q) \C(2,q)⊥ of weight w(c) ∈ [q + 2,2q− 1] satisfies21

c.` = α 6= 0, for some constant α valid for all lines ` of PG(2,q). Hence, supp(c) defines22

a blocking set of PG(2,q). This fact already makes the link to the article [5] on blocking23

sets. A detailed study of these possible codewords c shows that supp(c) must satisfy the24

following property: supp(c) must share 1 (mod p) points with every small linear blocking25

set of PG(2,q); see e.g. [5, Section 4], for the definition of linear blocking sets. Imposing26

this condition on supp(c) leads to the proof that supp(c) is equal to a line m of PG(2,q),27

but then supp(c) has weight q+1, which contradicts the fact that supp(c) ∈ [q+2,2q−1].28

This eliminates the existence of codewords of weight in [q+2,2q−1] in C(2,q)\C(2,q)⊥.29

Gathering all results gives that the second smallest weight of C(2,q) is equal to 2q.30

So here, geometrical ideas again lead to results on linear codes.31

For q = p prime, the following results are valid.32

Theorem 7.4. (Fack et al. [27]) The only codewords c, with 0 < w(c)≤ 2p+(p−1)/2, in33

the p-ary linear code C(2, p), p prime, p ≥ 11, are:34

• codewords with weight p+1: the scalar multiples of the incidence vectors of the lines35

of PG(2, p),36

• codewords with weight 2p: α(c1−c2), c1 and c2 the incidence vectors of two distinct37

lines of PG(2, p), α ∈ Fp \{0},38

ii


ii

ii

Uncorr

ected

Proof


• codewords with weight 2p + 1: αc1 + βc2, α,β ∈ Fp \ {0}, β 6= −α, with c1 and c21

the incidence vectors of two distinct lines of PG(2, p).2

The general results on the second smallest weight of the codes C(N,q) are as follows.3

Theorem 7.5. (Lavrauw et al. [56]) There are no codewords of weight in the interval [vN +4

1,2qN−1−1] in the code C(N,q).5

8 A geometrical result obtained via linear codes6

To conclude this article, we also wish to give a geometrical result obtained via coding-7

theoretical arguments.8

Definition 8.1. A strong representative system S in PG(N,q) is a set of points such that9

every point P ∈ S belongs to at least one tangent hyperplane TP(S) to S, i.e., every point P10

of S belongs to at least one hyperplane TP(S) only sharing P with S.11

The (q + 1)-arcs of PG(2,q) and the ovoids of PG(3,q) (see e.g. [24]) are particular12

examples of strong representative systems. Moreover, results of Bruen and Thas [10, 11]13

show that q√

q+1 is the largest size for a strong representative system in PG(2,q), q square,14

and that q2 +1 is the largest size for a strong representative system in PG(3,q). For N ≥ 4,15

Bruen and Thas prove that the size of every strong representative system S satisfies the16

bound |S|< q(N−1)/2. But the linear codes defined by the incidence matrices of points and17

hyperplanes of PG(N,q) lead to great improvements for large dimensions N.18

Techniques 8.2. The idea of obtaining an upper bound on the size of strong representative19

systems in PG(N,q) via linear codes is as follows.20

Every point P of S has a tangent hyperplane TP(S). Enumerate the points P1, . . . ,PvN+121

of PG(N,q) such that S = {P1, . . . ,P|S|}. Select for every point Pi of S a particular tangent22

hyperplane TPi(S), and enumerate the hyperplanes π1, . . . ,πvN+1 such that πi = TPi(S), i =23

1, . . . , |S|. Then the incidence matrix A of PG(N,q) has the following form:24

A =(

I|S| BC D

),25

with I|S| the identity matrix of rank |S|.26

This implies that rank(A)≥ |S|, but rank(A) is known (Subsection 7.1), so we find that27

the size of a strong representative system in PG(N,q) must satisfy28 (p+N−1

N

)h

+1 ≥ |S|.29

For large dimensions N, depending on the characteristic p, this upper bound on |S|30

improves greatly on the upper bound q(N−1)/2 found via the standard counting arguments.31

ii


ii

ii

Uncorr

ected

Proof


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ii


ii

ii

Uncorr

ected

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