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