introduction new tools in computer package for coding theory research and studying qplus are...

INTRODUCTION

New tools in computer package for coding theory research and studying QPlus are presented QPlus includes a DLL library package that implements coding theory algorithms

Methods for searching bounds on the size of q-ary equidistant codes by computer method

Examples for optimal equidistant codes and constant-weight equidistant codes found with QPlus

PRELIMINARIES

Eq(n, M, d) - equidistant code, consists of M vectors of length n over alphabet of q elements such that any two codewords differ in d positions

Eq(n, M, d, w) - constant-weight equidistant code - all the codewords have the same Hamming weight w

Bq(n,d), Bq(n,d,w) – the largest possible value of M when the other parameters are fixed Codes with such parameters are called optimal

One of the main open problem of algebraic coding theory is optimal codes searching

PRELIMINARIES

Lexicographic codes - of length n and Hamming distance d are obtained

by considering all q-ary vectors with weight w in lexicographic order, and

adding them to the code if they are at a distance exactly d from the words

that have been added earlier.

HISTORY

N. Semakov, V.Zinoviev, G. Zaitsev – equidistant codes and designs (1968, 1969)

J.I.Hall – bounds on equidistant codes (1977)

F.W.Fu, T.Klove, Y. Luo, V.K. Wei – upper bounds for constant-weight codes (2003)

G. Bogdanova, V. Zinoviev, T. Todorov – construction of q-ary equidistant codes (2007)

G. O.H. Katona, G. Bogdanova - equidistant codes for q=3, d=3 (2008)

HISTORY

GUAVA – linear codes

GFQ - calculations over finite fields

LinCoR - studying of binary linear codes

QLC - studying q-ary linear codes

QCC - searching of q-ary constant-weight codes from other codes

Q-Extension - linear codes researching, code equivalence etc.

EQUIDISTANT CODES SEARCHING

Fix first two codewords of the searched code with:

Codeword representation for codeword x:

Vector space – all codewords in lexicographic order that are on distance d

from the two fixed codewords

EQUIDISTANT CODES SEARCHING

Perform a backtrack search

• the distance between all the codewords in the code remains equal to d

• the newly added codeword doesn't break lexicographic order of the

columns in the code

Two columns b={b1, b2, ..., bM} and c={c1, c2, ..., cM} (b precedes c in the

code matrix) of a code have good lexicographic order of columns if bi=ci,

i=1...k, k≤ M and bk+1ck+1

EQUIDISTANT CODES SEARCHING

If we reach the end of the space we check if the size of the newly founded

code is bigger than the best code that we have up to this moment. If yes

then the newly founded code becomes best code. Finally we are doing a

step back and change the codewords on the previous levels.

LEXICOGRAPHIC CONSTANT-WEIGHT EQUIDISTANT CODE SEARCHING

Fix codewords that are included in the seed

Remove all the codewords that don't have weight w

Perform a greedy search - no backtracking.

If we reach the end of the space we output the code founded

LEXICOGRAPHIC CONSTANT-WEIGHT EQUIDISTANT CODE SEARCHING

The algorithm has the following options:

• Automatic search with each of the possible seeds with given size and

found the best code

• Search with cycle shift from the initial space which appears to produce

better codes then the standard space order

SOME RESULTS OBTAINED BY QPLUS

Backtrack search construction E4(4,9,3)

Construction with extension E5(6,25,5)

0 0 0 00 1 1 10 2 2 21 0 1 21 1 2 01 2 0 13 0 2 13 1 0 23 2 1 0

0 0 0 0 0 00 1 1 1 1 10 2 2 2 2 20 3 3 3 3 30 4 4 4 4 41 0 1 2 3 41 1 2 3 4 01 2 3 4 0 11 3 4 0 1 21 4 0 1 2 32 0 2 4 1 32 1 3 0 2 42 2 4 1 3 02 3 0 2 4 12 4 1 3 0 23 0 3 1 4 23 1 4 2 0 33 2 0 3 1 43 3 1 4 2 03 4 2 0 3 14 0 4 3 2 14 1 0 4 3 24 2 1 0 4 34 3 2 1 0 44 4 3 2 1 0

0 0 0 00 1 1 1

0 0 0 0 01 1 1 1 12 2 2 2 23 3 3 3 34 4 4 4 4

SOME RESULTS OBTAINED BY QPLUS

SOME RESULTS OBTAINED BY QPLUS

Construction of lexicographic codeE4(6,9,5)

0 0 0 0 0 01 0 1 1 1 10 1 1 2 2 20 2 2 1 3 31 1 3 3 0 32 2 0 3 1 22 3 2 2 0 13 2 3 0 2 13 3 1 3 3 0

1 0 1 1 1 1

SOME RESULTS OBTAINED BY QPLUS

MODULES OF QPLUS Application has modules for modular arithmetic, elementary number

theory, vectors and matrices arithmetic, linear codes researching

We add modules for equidistant, constant-weight equidistant and

lexicographic equidistant codes construction

The application has been successfully used for research and educational

purposes

We use Delphi's ActiveForm technology to create a Web-based version of

QPlus that offers most of the functionalities in web space

THANK YOU!