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Some Computation Problems in

Coding Theory

Eric Chen

Computer Science GroupHKr

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Outline

• Information Transmission System

• Some definitions

• Goals of coding theory

• Computer search for QT codes and QT 2-weight codes

• Some computation problems

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Information Transmission System• Source encoding (remove redundancy)• Channel encoder ( add redundancy )• Channel decoder ( error detection/correction)• Source decoding

Information Sink

Receiver(Decoder)

Transmitter(Encoder)

CommunicationChannel

Information Source

Noisek-digit k-digitn-digit n-digit

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Source coding and channel coding• Coding theory

The study of methods for efficient and reliable data transmission

• Source codingRemove redundancy (data compression)

• Channel codingAdd redundancy for error

detection/correction

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• 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 nEncoded as

codeword

Channel Coding – Principle

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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

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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}

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Definitions

• (Hamming) weight of a word xwt (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 = 110 (also called vectors)wt(v1)=2; wt(v2) =2d (v1, v2) = 2

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Definitions

• (Minimum) distance d of a code:The minimum distance between its

codewordsd = min weight of the non-zero codewords

• Minimum distance d determines the error detection or correction capabilityDetection d – 1 errorsCorrect (d – 1)/2 errors

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Example– repetition code

• Binary [m, 1, m] codek = 1, d = m

• Example [3, 1, 3] codeTwo codewords: 000, 111Each 0 000 Each 1 111Can detect 2 errors (for error detection)Can correct 1 error (for error correction)

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Example– repetition code

• Example [3, 1, 3] codeTwo codewords: 000, 1110 000 1 111Generator matrix

• G = [1 1 1]

Parity check matrix

101

011H

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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

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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

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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.

• NotationAq(n, d) is the largest M such that there

is an (n,M,d)q code.

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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

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Online Code Table

at http://www.codetables.de/

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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 = 20Bound on distance is 26 – 30

Difficult to improve the boundQuasi-twisted codes proven to contain good

or optimal codes.

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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) ??

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Online Code Tables

• Bounds for binary constant weight

codes

http://www.win.tue.nl/~aeb/codes/Andw.html

• Erik Agrell's tables of binary block

codes

http://webfiles.portal.chalmers.se/s2/research/kit/bou

nds/

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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

)(

)(

)(

1 xgx

xxg

xg

G

k

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Circulant Matrix• An nn 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

021

312

201

110

aaa

aaa

aaa

aaa

A nnn

nn

n

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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 – .

)(

)(

)(

1 xgx

xxg

xg

G

k

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Twistulant Matrix• An nn 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 - .

021

312

201

110

aaa

aaa

aaa

aaa

A nnn

nn

n

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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 codesevery consta-cyclic shift of a codeword by p

positions results in another codewordCalled quasi-twisted (QT) code

• QC code is a special case of QT code with = 1.

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Quasi-Twisted Code• Generator matrix  of QT [ pm, k] code

where Gij is a twistulant matrix of order m

• t-generator QT code1-generator QT codes have been well studied

tptt

p

p

GGG

GGG

GGG

G

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22221

11211

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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.

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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.

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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

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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 26271• To construct a QT [60, 19] code, it is

required to select 3 polynomials among 26271 Polynomials. The total number of combinations is 3 021 544 309 455.

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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 = 2 084 705

• of 3 021 544 309 455 – So a binary QC [60, 19, 18] code was constructed.

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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

tptt

p

p

GGG

GGG

GGG

G

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22221

11211

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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 codeA 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.

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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

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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

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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.

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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 coden = 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.

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Consta-Cyclic Simplex [21, 3]4 Code

twistulant generator matrix

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Quasi-Twisted Simplex [21, 3]4 Code

QT form of generator matrix

.

1001001101

1100001011

01110001011

1100100110

11110000101

1011100010

0111001001

0111100001

0101110001

0111001001

10111100001

11010111000

1011100100

1011110000

111010111000

0010111001

0010111100

001110101110

000101110

10001011110

010011101011

A

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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

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Weight Matrix

• Weight matrix for A(x) It is cyclic

• Example

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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.

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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.

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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

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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

0321

3012

2101

1210

...

...

...

...

...

dddd

dddd

dddd

dddd

D sss

ss

s

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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

2213323

3221332

2322133

3232213

3323221

1332322

2133232

D

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Online Database on Codes

• A web database of binary quasi-cyclic codes

http://moodle.tec.hkr.se/~chen/research/codes/searchqc2.htm

see also: codetables http://www.codetables.de

• A Web database of two-weight codeshttp://moodle.tec.hkr.se/~chen/research/2-weight-

codes/search.php

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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

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Computation Problems

• Computer search for QT codes1-generator QT codes2-generator QT codest-generator QT codes from QT simplex codes

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Computation Problems

• Computer search for QT 2-weight codesQT 2-weight codes from QT simplex codes

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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 values011000010100000000010001100001010000000001100110000101000000000010011000010100000000001001100001010000000000100110000101000000000010011000010100000000001001100001010000000000100110000101000000000010011000010100000000001001100001010000000000100110000101100000000010011000010010000000001001100001101000000000100110000010100000000010011000001010000000001001100000101000000000100110000010100000000010011100001010000000001001110000101000000000100