2. 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

3. PRELIMINARIES  E q (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  E q (n, M, d, w) - constant-weight equidistant code - all the codewords have the same Hamming weight w  B q (n,d), B q (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

4. 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.

5. 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)

6. 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.

7. 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

8. 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={b 1, b 2,..., b M } and c={c 1, c 2,..., c M } (b precedes c in the code matrix) of a code have good lexicographic order of columns if b i =c i, i=1...k, k≤ M and b k+1  c k+1

9. 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.

10. 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

11. 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

12. SOME RESULTS OBTAINED BY QPLUS Backtrack search construction E 4 (4,9,3) Construction with extension E 5 (6,25,5) 0 0 0 1 1 1 0 2 2 2 1 0 1 2 1 1 2 0 1 2 0 1 3 0 2 1 3 1 0 2 3 2 1 0 0 0 0 0 1 1 1 1 1 0 2 2 2 2 2 0 3 3 3 3 3 0 4 4 4 4 4 1 0 1 2 3 4 1 1 2 3 4 0 1 2 3 4 0 1 1 3 4 0 1 2 1 4 0 1 2 3 2 0 2 4 1 3 2 1 3 0 2 4 2 2 4 1 3 0 2 3 0 2 4 1 2 4 1 3 0 2 3 0 3 1 4 2 3 1 4 2 0 3 3 2 0 3 1 4 3 3 1 4 2 0 3 4 2 0 3 1 4 0 4 3 2 1 4 1 0 4 3 2 4 2 1 0 4 3 4 3 2 1 0 4 4 4 3 2 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4

13. SOME RESULTS OBTAINED BY QPLUS

14. Construction of lexicographic code E 4 (6,9,5) 0 0 0 1 0 1 1 1 1 0 1 1 2 2 2 0 2 2 1 3 3 1 1 3 3 0 3 2 2 0 3 1 2 2 3 2 2 0 1 3 2 3 0 2 1 3 3 1 3 3 0 1 0 1 1 1 11 0 1 1 1 1

15. SOME RESULTS OBTAINED BY QPLUS

16. 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

17. THANK YOU!