reed solomon codes over fields of characteristic zero · 2019-07-22 · ulm university institute of...
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![Page 1: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/1.jpg)
Ulm University Institute of Communications Engineering
Reed–Solomon Codesover
Fields of Characteristic Zero
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert
ISIT 2019, Paris
A NACHRICHTENTECHNIK
July 10, 2019
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 1
![Page 2: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/2.jpg)
Ulm University Institute of Communications Engineering
Motivation A NACHRICHTENTECHNIK
We know Reed–Solomon Codes over
Fq CElements are represented witha fixed number of bits
Operations cost a constantnumber of bit operations
Floating point operations areused
Problem: Rounding errors
Aim
Reed–Solomon Codes over arbitrary fields with exact calculations duringEncoding and Decoding.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 2
![Page 3: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/3.jpg)
Ulm University Institute of Communications Engineering
Motivation A NACHRICHTENTECHNIK
We know Reed–Solomon Codes over
Fq CElements are represented witha fixed number of bits
Operations cost a constantnumber of bit operations
Floating point operations areused
Problem: Rounding errors
Aim
Reed–Solomon Codes over arbitrary fields with exact calculations duringEncoding and Decoding.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 2
![Page 4: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/4.jpg)
Ulm University Institute of Communications Engineering
GRS Codes over arbitrary Fields A NACHRICHTENTECHNIK
Definition: Generalization of Definition 5.1.1 in [Rot06]
Let K be a field and k, n ∈ N such that k ≤ n. Chooseα1, . . . , αn ∈ K\0 to be distinct and v1, . . . , vn ∈ K \ 0. Wedefine the generalized Reed–Solomon Code CGRS ⊆ Kn with paritycheck matrix
HVandermonde =
1 1 . . . 1α1 α2 . . . αn...
... . . ....
αn−k−11 αn−k−1
2 . . . αn−k−1n
v1v2
. . .
vn
.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 3
![Page 5: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/5.jpg)
Ulm University Institute of Communications Engineering
GRS Codes over arbitrary Fields A NACHRICHTENTECHNIK
Generator Matrix
A generator matrix is of the form
GVandermonde =
1 1 . . . 1α1 α2 . . . αn...
... . . ....
αk−11 αk−1
2 . . . αk−1n
v′1
v′2. . .
v′n
,
where the v′i ∈ K \ 0, given by the following linear system ofequations:
n∑i=1
αri viv
′i = 0 ∀ r = 0, . . . , n− 2 .
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 4
![Page 6: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/6.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth A NACHRICHTENTECHNIK
over Fields of Characteristic Zero
If the underlying field is of characteristic zero the coefficientsduring Encoding and Decoding will grow.
Example: Euclidean Algorithm
f0, f1 ∈ F1789[x] g0, g1 ∈ Q[t]
f0(x) = x10 − 3
f1(x) = 3x9 − 2
g0(t) = t10 − 3
g1(t) = 3t9 − 2
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 5
![Page 7: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/7.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth A NACHRICHTENTECHNIK
over Fields of Characteristic Zero
Example: Euclidean Algorithm - Step 1
(x10 − 3)/(3x9 − 2)
= 1193x
Remainder: 597x+ 1786
(t10 − 3)/(3t9 − 2)
=1
3t
Remainder:2
3t− 3
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 6
![Page 8: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/8.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth A NACHRICHTENTECHNIK
over Fields of Characteristic Zero
Example: Euclidean Algorithm - Step 2
(3x9 − 2)/(597x+ 1786)
= 899x8 + 1362x7 + 762x6
+ 1640x5 + 224x4 + 1008x3
+ 958x2 + 733x+ 615
Remainder: 54
(3t9 − 2)/(2
3t− 3)
=9
2t8 +
81
4t7 +
729
8t6 +
6561
16t5
+59049
32t4 +
531441
64t3
+4782969
128t2 +
43046721
256t
+387420489
512
Remainder:1162260443
512
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 7
![Page 9: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/9.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth A NACHRICHTENTECHNIK
over Fields of Characteristic Zero
Example: Euclidean Algorithm - Step 3
(597x+ 1786)/54
= 508x+ 1292
Remainder: 0
(2
3t− 3)/
1162260443
512
=1024
3486781329t− 1536
1162260443Remainder: 0
Question:
Is it possible to derive bounds for the growth of the coefficientsduring Encoding and Decoding?
→ Solution with the help of already known results from computeralgebra.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 8
![Page 10: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/10.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth A NACHRICHTENTECHNIK
over Fields of Characteristic Zero
Example: Euclidean Algorithm - Step 3
(597x+ 1786)/54
= 508x+ 1292
Remainder: 0
(2
3t− 3)/
1162260443
512
=1024
3486781329t− 1536
1162260443Remainder: 0
Question:
Is it possible to derive bounds for the growth of the coefficientsduring Encoding and Decoding?
→ Solution with the help of already known results from computeralgebra.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 8
![Page 11: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/11.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth A NACHRICHTENTECHNIK
over Fields of Characteristic Zero
Example: Euclidean Algorithm - Step 3
(597x+ 1786)/54
= 508x+ 1292
Remainder: 0
(2
3t− 3)/
1162260443
512
=1024
3486781329t− 1536
1162260443Remainder: 0
Question:
Is it possible to derive bounds for the growth of the coefficientsduring Encoding and Decoding?
→ Solution with the help of already known results from computeralgebra.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 8
![Page 12: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/12.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
We define the bit width λ(a): (Generalization of [vzGG13] p. 142)
a ∈ Z:
λ(a) :=
blog2(|a|)c+ 1, if a 6= 0
0, if a = 0
a = bc ∈ Q with b, c ∈ Z, c 6= 0, and gcd(b, c) = 1:
λ(a) := maxλ(b), λ(c).
a(x) =∑r
i=0aib · x
i ∈ Q[x] with ai ∈ Z and b ∈ N \ 0 suchthat gcd(a0, . . . , ar, b) = 1:
λ(a(x)) := maxλ(a0), . . . , λ(ar), λ(b).
NEW: A = (aij) ∈ Qk×r:
λ(A) = maxλ(aij) : i = 1, . . . , k and j = 1, . . . , r.Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9
![Page 13: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/13.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
We define the bit width λ(a): (Generalization of [vzGG13] p. 142)
a ∈ Z:
λ(a) :=
blog2(|a|)c+ 1, if a 6= 0
0, if a = 0
a = bc ∈ Q with b, c ∈ Z, c 6= 0, and gcd(b, c) = 1:
λ(a) := maxλ(b), λ(c).
a(x) =∑r
i=0aib · x
i ∈ Q[x] with ai ∈ Z and b ∈ N \ 0 suchthat gcd(a0, . . . , ar, b) = 1:
λ(a(x)) := maxλ(a0), . . . , λ(ar), λ(b).
NEW: A = (aij) ∈ Qk×r:
λ(A) = maxλ(aij) : i = 1, . . . , k and j = 1, . . . , r.Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9
![Page 14: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/14.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
We define the bit width λ(a): (Generalization of [vzGG13] p. 142)
a ∈ Z:
λ(a) :=
blog2(|a|)c+ 1, if a 6= 0
0, if a = 0
a = bc ∈ Q with b, c ∈ Z, c 6= 0, and gcd(b, c) = 1:
λ(a) := maxλ(b), λ(c).
a(x) =∑r
i=0aib · x
i ∈ Q[x] with ai ∈ Z and b ∈ N \ 0 suchthat gcd(a0, . . . , ar, b) = 1:
λ(a(x)) := maxλ(a0), . . . , λ(ar), λ(b).
NEW: A = (aij) ∈ Qk×r:
λ(A) = maxλ(aij) : i = 1, . . . , k and j = 1, . . . , r.Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9
![Page 15: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/15.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
We define the bit width λ(a): (Generalization of [vzGG13] p. 142)
a ∈ Z:
λ(a) :=
blog2(|a|)c+ 1, if a 6= 0
0, if a = 0
a = bc ∈ Q with b, c ∈ Z, c 6= 0, and gcd(b, c) = 1:
λ(a) := maxλ(b), λ(c).
a(x) =∑r
i=0aib · x
i ∈ Q[x] with ai ∈ Z and b ∈ N \ 0 suchthat gcd(a0, . . . , ar, b) = 1:
λ(a(x)) := maxλ(a0), . . . , λ(ar), λ(b).
NEW: A = (aij) ∈ Qk×r:
λ(A) = maxλ(aij) : i = 1, . . . , k and j = 1, . . . , r.Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9
![Page 16: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/16.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
Examples
(i) λ(127) = blog2(|127|)c+ 1 = 7
(ii) λ( 364) = max λ(3)︸︷︷︸
blog2(|3|)c+1
, λ(64)︸ ︷︷ ︸blog2(|64|)c+1
= max1, 7 = 7
(iii) λ(2x3 + 25x
2 + 18) = λ(80x
3+16x2+540 )
= maxλ(80), λ(16), λ(5), λ(40)= λ(80) = blog2(|80|)c+ 1 = 7
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 10
![Page 17: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/17.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
Examples
(i) λ(127) = blog2(|127|)c+ 1 = 7
(ii) λ( 364) = max λ(3)︸︷︷︸
blog2(|3|)c+1
, λ(64)︸ ︷︷ ︸blog2(|64|)c+1
= max1, 7 = 7
(iii) λ(2x3 + 25x
2 + 18) = λ(80x
3+16x2+540 )
= maxλ(80), λ(16), λ(5), λ(40)= λ(80) = blog2(|80|)c+ 1 = 7
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 10
![Page 18: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/18.jpg)
Ulm University Institute of Communications Engineering
The bit width - a Measure of Coefficient GrowthA NACHRICHTENTECHNIK
Examples
(i) λ(127) = blog2(|127|)c+ 1 = 7
(ii) λ( 364) = max λ(3)︸︷︷︸
blog2(|3|)c+1
, λ(64)︸ ︷︷ ︸blog2(|64|)c+1
= max1, 7 = 7
(iii) λ(2x3 + 25x
2 + 18) = λ(80x
3+16x2+540 )
= maxλ(80), λ(16), λ(5), λ(40)= λ(80) = blog2(|80|)c+ 1 = 7
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 10
![Page 19: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/19.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Encoding A NACHRICHTENTECHNIK
over the Rational Numbers
Bound for the bit width of the codeword
Let c be an RS codeword generated by encoding u ∈ Qk withgenerator matrix G ∈ Qk×n. Then
λ(c) ≤ k(λ(u) + λ(G) + 1).
Generator Matrix in systematic form [RS85, Theorem 1]
CGRS has a systematic generator matrix of the form
Gsys = (Ik×k | A), where A =(
cidjai−bj
)is a Cauchy matrix with
ai, bj , ci, dj dependent on αi andv′i ∀i = 1, . . . , k, j = 1, . . . , n− k.
→ λ(G) small
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 11
![Page 20: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/20.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Encoding A NACHRICHTENTECHNIK
over the Rational Numbers
Bound for the bit width of the codeword
Let c be an RS codeword generated by encoding u ∈ Qk withgenerator matrix G ∈ Qk×n. Then
λ(c) ≤ k(λ(u) + λ(G) + 1).
Generator Matrix in systematic form [RS85, Theorem 1]
CGRS has a systematic generator matrix of the form
Gsys = (Ik×k | A), where A =(
cidjai−bj
)is a Cauchy matrix with
ai, bj , ci, dj dependent on αi andv′i ∀i = 1, . . . , k, j = 1, . . . , n− k.
→ λ(G) small
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 11
![Page 21: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/21.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Encoding A NACHRICHTENTECHNIK
over the Rational Numbers
Bound for the bit width of the codeword
Let c be an RS codeword generated by encoding u ∈ Qk withgenerator matrix G ∈ Qk×n. Then
λ(c) ≤ k(λ(u) + λ(G) + 1).
Generator Matrix in systematic form [RS85, Theorem 1]
CGRS has a systematic generator matrix of the form
Gsys = (Ik×k | A), where A =(
cidjai−bj
)is a Cauchy matrix with
ai, bj , ci, dj dependent on αi andv′i ∀i = 1, . . . , k, j = 1, . . . , n− k.→ λ(G) small
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 11
![Page 22: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/22.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Encoding A NACHRICHTENTECHNIK
over the Rational Numbers
Bound for the bit width of the codeword
Let c be an RS codeword generated by encoding u ∈ Qk withgenerator matrix G ∈ Qk×n. Then
λ(c) ≤ k(λ(u) + λ(G) + 1).
Generator Matrix in systematic form [RS85, Theorem 1]
CGRS has a systematic generator matrix of the form
Gsys = (Ik×k | A), where A =(
cidjai−bj
)is a Cauchy matrix with
ai, bj , ci, dj dependent on αi andv′i ∀i = 1, . . . , k, j = 1, . . . , n− k.→ λ(G) small
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 11
![Page 23: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/23.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Encoding A NACHRICHTENTECHNIK
over the Rational Numbers
Comparison of systematic and non-systematic Encoding
For a special choice of v′ we get λ(Gsys) < λ(GVandermonde)Upper Bounds for the bit width of the Generatormatix
λ(GVandermonde) λ(Gsys)
general (k − 1)λ(α) + λ(v′) 2(2k − 1)λ(α) + 2λ(v′)+2k − 1
cidj = 1 (k − 1)(3λ(α) + 1) 2λ(α) + 1
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 12
![Page 24: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/24.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Encoding A NACHRICHTENTECHNIK
over the Rational Numbers
0 1,000 2,000 3,000 4,0000
0.5
1
1.5
·104
n
aver
age
bit
wid
thofc
α = α1, GV, v′i = 1α = α1, Gsys, cidj = 1
α = α2, GV, v′i = 1α = α2, Gsys, cidj = 1
α1 := (1, 2, . . . , n)
α2 := (−1, 1,− 12,
12,−2, 2, 1
3,− 1
3,
23, . . . )
We chose 1000information words ofbit width 100.
Rate: k = bn/3c.
λ(GVandermonde) λ(Gsys)
general (k − 1)λ(α) + λ(v′) 2(2k − 1)λ(α) + 2λ(v′)+2k − 1
cidj = 1 (k − 1)(3λ(α) + 1) 2λ(α) + 1
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 13
![Page 25: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/25.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Decoding A NACHRICHTENTECHNIK
over the Rational Numbers (Generalization of [Rot06] Chapter 6)
Algorithm 1: Decoding Algorithm for GRS Codes over Q
Input: Received Word r = c+ e, where c ∈ CGRS and wtH(e) ≤ bn−k2 c.Output: Codeword c
1 s← rH>Vandermonde
2 S(x)←∑d−2i=0 six
i
3 ξ ← lcm(den(s0), . . . ,den(sd−2))
4 (rh, sh, th)← EEA(ξ · xd−1, ξ · S(x), d−12 ) // implementation of
[vzGG13, Algorithm 6.57]
5 c← 0th coefficient of th6 (Λ(x),Ω(x))← c−1 · (th, rhξ )
7 Λ′(x)←∑i>0 iΛix
i−1
8 ei ← −αi
vi
Ω(α−1i )
Λ′(α−1i )
for i = 1, . . . , n
9 return c = r − e
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 14
![Page 26: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/26.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Decoding A NACHRICHTENTECHNIK
over the Rational Numbers
Bit width of the Syndrome
Let r = c+ e be a received word, s = rH>Vandermonde thesyndrome and τ = wtH(e). For the bit width of s we get thefollowing bound:
λ(s) ≤ τ(λ(e) + λ(HVandermonde) + 1).
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 15
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Ulm University Institute of Communications Engineering
Coefficient Growth in Decoding A NACHRICHTENTECHNIK
over the Rational Numbers
Complexity of the Algorithm
The complexity in bit operations is
O∼(d7[λ(e) + λ(HGRS)
]2+ n4[λ(c) + λ(e) + λ(HGRS)]
).
If the error e has bit width at most t, codeword c at most t′
and α is chosen choosen such that λ(α) ∈ O(log(n)) (e.g.,α1 or α2) then Algorithm 1 can be implemented in
O∼(maxn7t2, n9, n4t′
)bit operations.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 16
![Page 28: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/28.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Decoding A NACHRICHTENTECHNIK
over the Rational Numbers
Complexity of the Algorithm
The complexity in bit operations is
O∼(d7[λ(e) + λ(HGRS)
]2+ n4[λ(c) + λ(e) + λ(HGRS)]
).
If the error e has bit width at most t, codeword c at most t′
and α is chosen choosen such that λ(α) ∈ O(log(n)) (e.g.,α1 or α2) then Algorithm 1 can be implemented in
O∼(maxn7t2, n9, n4t′
)bit operations.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 16
![Page 29: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/29.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Decoding A NACHRICHTENTECHNIK
Comparison of the maximum bit width λmax for Decoding using different Variants of theEEA
20 25 30 35 40 45 500
100
200
300
400
500
600
code length n
max
imu
mb
itw
idthλmax
Algorithm 1implementation with usual EEA
We chose λ(e) = 40, d = 11For each point 100 simulations were carried out
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 17
![Page 30: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/30.jpg)
Ulm University Institute of Communications Engineering
Coefficient Growth in Decoding A NACHRICHTENTECHNIK
Comparison of the maximum bit width λmax for Decoding using different Variants of theEEA
4 6 8 10 12 14 16 18 20 220
200
400
600
800
1,000
minimum distance d
max
imu
mb
itw
idthλmax
Algorithm 1implementation with usual EEA
We chose λ(e) = 40 and n = 40For each point 100 simulations were carried out
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 18
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Ulm University Institute of Communications Engineering
Conclusion A NACHRICHTENTECHNIK
Properties of Reed–Solomon Codes over Fq also hold overarbitrary fields
Over Q there exist bounds for the coefficient growth duringencoding
Over Q decoding up to half-the-minimum distance is possiblein a polynomial number of bit operations
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 19
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Ulm University Institute of Communications Engineering
Conclusion A NACHRICHTENTECHNIK
Properties of Reed–Solomon Codes over Fq also hold overarbitrary fields
Over Q there exist bounds for the coefficient growth duringencoding
Over Q decoding up to half-the-minimum distance is possiblein a polynomial number of bit operations
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 19
![Page 33: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/33.jpg)
Ulm University Institute of Communications Engineering
Conclusion A NACHRICHTENTECHNIK
Properties of Reed–Solomon Codes over Fq also hold overarbitrary fields
Over Q there exist bounds for the coefficient growth duringencoding
Over Q decoding up to half-the-minimum distance is possiblein a polynomial number of bit operations
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 19
![Page 34: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/34.jpg)
Ulm University Institute of Communications Engineering
Future Work A NACHRICHTENTECHNIK
Extension of the results to more classes of number fields, forinstance Q[i].
Consider other decoding algorithms, e.g. Berlekamp-Welch,Berlekamp-Massey or list decoding approaches
Reduction of the computation modulo a prime bydecomposing the number field into prime ideals such as in[ALR17]
Determine the bit complexity of Decoding algorithms forGabidulin codes over characteristic zero with the samemethods.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 20
![Page 35: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/35.jpg)
Ulm University Institute of Communications Engineering
Future Work A NACHRICHTENTECHNIK
Extension of the results to more classes of number fields, forinstance Q[i].
Consider other decoding algorithms, e.g. Berlekamp-Welch,Berlekamp-Massey or list decoding approaches
Reduction of the computation modulo a prime bydecomposing the number field into prime ideals such as in[ALR17]
Determine the bit complexity of Decoding algorithms forGabidulin codes over characteristic zero with the samemethods.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 20
![Page 36: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/36.jpg)
Ulm University Institute of Communications Engineering
Future Work A NACHRICHTENTECHNIK
Extension of the results to more classes of number fields, forinstance Q[i].
Consider other decoding algorithms, e.g. Berlekamp-Welch,Berlekamp-Massey or list decoding approaches
Reduction of the computation modulo a prime bydecomposing the number field into prime ideals such as in[ALR17]
Determine the bit complexity of Decoding algorithms forGabidulin codes over characteristic zero with the samemethods.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 20
![Page 37: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/37.jpg)
Ulm University Institute of Communications Engineering
Future Work A NACHRICHTENTECHNIK
Extension of the results to more classes of number fields, forinstance Q[i].
Consider other decoding algorithms, e.g. Berlekamp-Welch,Berlekamp-Massey or list decoding approaches
Reduction of the computation modulo a prime bydecomposing the number field into prime ideals such as in[ALR17]
Determine the bit complexity of Decoding algorithms forGabidulin codes over characteristic zero with the samemethods.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 20
![Page 38: Reed Solomon Codes over Fields of Characteristic Zero · 2019-07-22 · Ulm University Institute of Communications Engineering Reed{Solomon Codes over Fields of Characteristic Zero](https://reader033.vdocument.in/reader033/viewer/2022043009/5f9ba254de1189223d44f107/html5/thumbnails/38.jpg)
Ulm University Institute of Communications Engineering
References A NACHRICHTENTECHNIK
[ALR17] Daniel Augot, Pierre Loidreau, and Gwezheneg Robert.Generalized Gabidulin Codes Over Fields of Any Characteristic.Des. Codes Cryptogr., pages 1–42, 2017.
[Rot06] Ron M. Roth.Introduction to Coding Theory.Cambridge UP, 2006.
[RS85] Ron M Roth and Gadiel Seroussi.On Generator Matrices of MDS Codes.IEEE Trans. Inf. Theory, 31(6):826–830, November 1985.
[SOPB19] Carmen Sippel, Cornelia Ott, Sven Puchinger, and Martin Bossert.Reed–Solomon Codes over Fields of Characteristic Zero, 2019.Available athttps://nt.uni-ulm.de/sippelottpuchrs2019extended.
[vzGG13] Joachim von zur Gathen and Jurgen Gerhard.Modern Computer Algebra.Cambridge university press, 2013.
Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 21