1. Some Computation Problems in Coding Theory
Eric Chen Computer Science Group HKr
Allmän föreläsning för alla anställda vid Högskolan med intresse för EU-forskning. Föreläsare Professor Claes Magnusson Claes Magnusson arbetar f.n. med utveckling av maskinteknikprogrammet, Kristianstad Högskola. Han har sedan tidigare erfarenhet av EU:s forskningsprogram, både som projektledare och utvärderare och han arbetade i tre år som ”dörröppnare” i Bryssel åt svenska universitet och högskolor. Claes kommer i sitt anförande behandla frågor som: • Vilka forskningsområden finns? • Hur gör man en ansökan? • Hur är det att vara med? • Hur går det för Sverige? • Hur lång tid tar det innan pengarna kommer? • Vilken hjälp finns att få? • På vilka sätt kan jag delta? • Vilka fördelar kan deltagandet ge?

2. Outline Information Transmission System Some definitions
Goals of coding theory
Computer search for QT codes and QT 2-weight codes
Some computation problems

3. Information Transmission System
Source encoding (remove redundancy)
Channel encoder ( add redundancy )
Channel decoder ( error detection/correction)
Source decoding
Information Sink
Receiver (Decoder)
Transmitter (Encoder)
Communication Channel
Information Source
Noise
n-digit
n-digit
k-digit
k-digit

4. Source coding and channel coding
Coding theory
The study of methods for efficient and reliable data transmission
Source coding
Remove redundancy (data compression)
Channel coding
Add redundancy for error detection/correction

5. Channel Coding – Principle
add additional information, or redundancy to data
added by sender, checked by receiver
k data digits encoded to a codeword of n digits
Code rate r = k / n k n
Encoded as
codeword

6. Binary linear block code
A block of k digits u = u1u2 … uk.
ui= 0 or 1 for a binary code
Encoded into a codeword x = x1x2…xn.
The mapping u  x
Generator matrix G k × n matrix
x = u G
Parity check matrix H (n-k) × n matrix
H xT = 0

7. Definitions bit (binary digit) : 0 or 1 Digit : an element of GF(q)
Word : a sequence of digits or bits
Binary code : a set of words over GF(2)
Example
C2 = {000, 011, 101, 110}

8. Definitions (Hamming) weight of a word x
wt (x) the number of non-zero digit in a word
(Hamming) distance between two words
the number of positions where they differ
Example
Words v1 = 101, v2 = (also called vectors)
wt(v1)=2; wt(v2) =2
d (v1, v2) = 2

9. Definitions (Minimum) distance d of a code:
The minimum distance between its codewords
d = min weight of the non-zero codewords
Minimum distance d determines the error detection or correction capability
Detection d – 1 errors
Correct (d – 1)/2 errors

10. Example– repetition code
Binary [m, 1, m] code
k = 1, d = m
Example [3, 1, 3] code
Two codewords: 000, 111
Each 0  000
Each 1  111
Can detect 2 errors (for error detection)
Can correct 1 error (for error correction)

11. Example– repetition code
Example [3, 1, 3] code
Two codewords: 000, 111
0   111
Generator matrix
G = [1 1 1]
Parity check matrix

12. 4 Foundamental Parameters of a Linear Code
Code dimension, k
Block length, n
Minimum distance, d
Alphabet size, q
A linear q-ary code is often written as an [n, k, d]q code
q=2, called binary code, [n, k, d] code
Code rate: r = k/n

13. 4 Foundamental Parameters of a Code
An (n, M, d)q code:
Number of the codewords, M
Block length, n
Minimum distance, d
Alphabet size, q q is a prime power
q=2, called binary code, (n, M, d) code
Code rate: r = log M / n

14. The Goals of Coding Theory
A good q-ary (n, M, d) code has small n, large M and large d.
The main coding theory problem is to optimize one of the parameters n, M, d for given values of the other two.
Notation
Aq(n, d) is the largest M such that there is an (n,M,d)q code.

15. The Goals of Coding Theory
For linear q-ary codes:
Given k, d, q. Find an [n, k, d]q code that minimizes n
Given n, d, q. Find an [n, k, d]q code that maximizes k
Given n, k, q. Find an [n, k, d]q code that maximizes d

16. Online Code Table at

17. Example Given n and k, maximize the distance ? n = 60, k = 19
the lower bound on distance is 18
the upper bound on distance is 20
if [60, 19] code exists with d = 19 or 20 ?
n = 81, k = 20
Bound on distance is 26 – 30
Difficult to improve the bound
Quasi-twisted codes proven to contain good or optimal codes.

18. Binary constant weight codes
All codewords have a weight w
A(n,d,w) is the maximum size of a binary code with word length n, minimum distance d, and constant weight w.
Closely related to combinatorial designs
How to determine the A(n, d, w) ??

19. Erik Agrell's tables of binary block codes
Online Code Tables
Bounds for binary constant weight codes
Erik Agrell's tables of binary block codes

20. Cyclic Code and Polynomial
Every cyclic shift of a codeword is also a codeword  
Generator polynomial g(x) and generator matrix
operation modulo xn – 1.
Many famous cyclic codes
BCH codes, Reed-Solomon codes

21. Circulant Matrix An nn cyclic or circulant matrix is defined as
it is uniquely specified by a polynomial formed by the elements of its first row, a(x) = a0 + a1 x + a2 x2 + … + an-1 xn-1
Operation modulo xn – 1

22. Consta-cyclic Code
Every consta-cyclic shift of a codeword is also a codeword  
(a0, a1,…, an-1)  (an-1 ,a0, a1,…, an-2)
Generator polynomial g(x) and generator matrix
operation modulo xn – .

23. Twistulant Matrix
An nn consta-cyclic or twistulant matrix is defined as
it is uniquely specified by a polynomial formed by the elements of its first row, a(x) = a0 + a1 x + a2 x2 + … + an-1 xn-1.
Operation modulo xn - .

24. Quasi-Cyclic Code A generalization of cyclic codes
every cyclic shift of a codeword by p positions results in another codeword
Called quasi-cyclic (QC) code
A generalization of consta-cyclic codes
every consta-cyclic shift of a codeword by p positions results in another codeword
Called quasi-twisted (QT) code
QC code is a special case of QT code with = 1.

25. Quasi-Twisted Code Generator matrix of QT [ pm, k] code
where Gij is a twistulant matrix of order m
t-generator QT code
1-generator QT codes have been well studied

26. 1-generator QT [mp, k] code G = [G0 G1 G2 … Gp-1 ]
1-generator QT codes
1-generator QT [mp, k] code
G = [G0 G1 G2 … Gp-1 ]
Let g0(x), g1(x), …, gp-1(x) be the defining polynomials  
k = m – degree( gcd(g0(x), g1(x), …, gp-1(x), xm – 1 ))
It is called de-generated if k < m.

27. 1-generator QT [mp, k] code G = [G0 G1 G2 … Gp-1 ]
Defining polynomials
1-generator QT [mp, k] code
G = [G0 G1 G2 … Gp-1 ]
defining polynomials: g0(x), g1(x), …, gp-1(x)  
(g0(x), g1(x), …, gp-1(x) ) and (axjg0(x), g1(x), …, gp-1(x)) defines the equivalent QT code.
a is any non-zero element in GF(q)
j = 1, 2, …, m -1.

28. Computer search for QT codes
1-generator QT [mp, k] code
G = [G0 G1 G2 … Gp-1 ]
Find all candidate polynomials
Equivalent classes defined by axjg(x)
Select p polynomials from non-equivalent polynomials

29. Computer search example for QT codes
Binary QC [60, 19, 18] code
m = 20, k = 19, previously best known d = 17
The number of non-equivalent polynomials is
To construct a QT [60, 19] code, it is required to select 3 polynomials among Polynomials. The total number of combinations is

30. Computer search example for QT codes
Binary QC [60, 19, 18] code
m = 20, k = 19. Previously best known d = 17
My paper in IEEE IT 1994
g0(x) = 1 + x
Divide the candidate polynomials into groups based on their weights
g1(x) and g2(x) are chosen from sets of polynomials with weights 4 and 12, respectively.
Total # of combinations: 245 X 8509 = of
So a binary QC [60, 19, 18] code was constructed.

31. t-generator QT codes t-generator QT [mp, k] code
Where G are twistulants of order m.
Most research has been focused on 1-generator QC or QT codes.
Computer search becomes more time-consuming

32. Two-Weight Codes
A [n, k] code is a two-weight code if any non-zero codeword has a weight of w1 or w2.
Notation: [n, k; w1, w2]q code
Projective code
A code is said to be projective if any two of its coordinates are linearly independent, or, if the minimum distance of its dual code is at least three.

33. Why studying 2-weight codes
Linear constant weight codes (simplex codes) are optimal  many 2-weight codes are also optimal
Related to strongly regular graphs

34. Simplex Codes Simplex [(qt–1)/(q–1), t]q code
equi-distance code, d = qt-1
All non-zero codewords have the same weight, d = qt-1
A λ-consta-cyclic simplex code can be defined by a generator polynomial g(x) = (xn–l)/h(x),
where n=(qt–1) /(q–1), and λ is a non-zero element of GF(q) and has order of q–1

35. QT form of a simplex code
If the block length n = (qt – 1)/(q-1) is not a prime, n = ms.
The simplex code can be put into a QT code with s blocks.
If only taking p blocks (among s), a QT code can be constructed.
QT codes, QT 2-weight codes can be constructed in this way.

36. QT Simplex Codes
If n=(qt–1) /(q–1) = mr, Simplex [(qt–1)/(q–1), t]q code can be put into QT from.
Example:simplex [21, 3]4 code
n = 21 = mp = 3 × 7, m = 3, p = r, q = 4.
Let 0, 1, a, and b = 1 + a be elements of GF(4),
λ=b. Then a λ-consta-cyclic matrix defined by c(x) = 1+ bx + bx3 + bx4 + bx5 + ax6 +x7 + x8 + ax9 + x10 + ax11 + x13 +ax15 +bx16 +x17 + x18.

37. Consta-Cyclic Simplex [21, 3]4 Code
twistulant generator matrix

38. Quasi-Twisted Simplex [21, 3]4 Code
QT form of generator matrix

39. Quasi-Twisted Simplex [21, 3]4 Code
QT form of generator matrix
Representation by polynomials
a1(x) = 1 +x, a2(x) = b + ax + x2 , a3(x) = ax + bx2 , a4(x) = b + x + x2, a5(x) = b + ax + x2, a6(x) = b, a7(x) = a+ x. r = 7

40. Weight Matrix
Weight matrix for A(x)
It is cyclic
Example

41. Computer Construction of QT 2-Weight Codes
Given a simplex [mr, t]q code of composite length n =(qt–1) /(q–1) = mr
Find the generator polynomial,
Obtain A(x) and weight matrix
To construct a QT 2-weight [mp, t; w1, w2] code, it is to find p columns such that the row sums of the selected columns give w1 or w2.

42. Computer Construction of QT 2-Weight Codes
Example
From simplex [21, 3]4 code with m=3
A QT 2-weight [9, 3; 6, 8]4 code can be constructed by columns 1, 2, and 4.

43. Results A large amount of QT 2-weight codes have been obtained.
Most codes have the same parameters as known codes.
They may not be equivalent
Exmaple [154, 6; 99, 108]3 code
Gulliver constructed with m = 11, p =14
Using the method above, m = 7, p =22
They are not equivalent
Some new codes are obtained

44. Computer search for QT Codes
Given a cyclic weight matrix of order s
How to select p columns such that
Maximize the minimum row sums of p cols

45. Computer search for QT codes
Given a cyclic weight matrix of order s
The row sums for columns 0, 1, 3 are 8, 6, 6, 6, 8, 6, and 8, respectively
Taking columns 0, 1, and 3  a QT [9, 3, 6]3 code

46. Online Database on Codes
A web database of binary quasi-cyclic codes
see also: codetables
A Web database of two-weight codes

47. Computation Problems Improve lower bound on A(n, w, d)
Improve lower bound on distance for given n, k, q
Computer search for 2-weight codes

48. Computation Problems Computer search for QT codes 1-generator QT codes
t-generator QT codes from QT simplex codes

49. Computer search for QT 2-weight codes
Computation Problems
Computer search for QT 2-weight codes
QT 2-weight codes from QT simplex codes

50. Computation Problems Computer search for QT 2-weight codes
Study on a binary cyclic matrix A
Select p columns such that the corresponding row sums are of two values