Abstract
The main problem of coding theory is to construct codes with large Hamming-distances between the code-words. In this work we describe a fast algorithm for generating pairs of q-ary codes with prescribed pairwise Hamming-distances and coincidences (for a letter s ∈ {0,1,...,q − 1}, the number of s-coincidences between codewords a and b is the number of letters s in the same positions both in a and b). The method is a generalization of a method for constructing set-systems with prescribed intersection sizes (Grolmusz (2002) Constructing set-systems with prescribed intersection sizes. J Algorithms 44:321–337), where only the case q = 2 and s = 1 was examined. As an application, we show that the modular version of the classical Delsarte-inequality does not hold for odd, non-prime power composite moduli.
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L Babai H Snevily RM Wilson (1995) ArticleTitleA new proof for several inequalities on codes and sets J Combinator Theory A 71 146–153 Occurrence Handle0826.05053 Occurrence Handle1335782 Occurrence Handle10.1016/0097-3165(95)90021-7
Mix Barrington DA, Beigel R, Rudich S (1994/1992) Representing Boolean functions as polynomials modulo composite numbers. Comput Complexity 4:367–382, 1994. Appeared also in Proc 24th Ann ACM Symp Theor Comput 1992.
Delsarte P (1973) An algebraic approach to the association schemes of coding theory. Philips Res Rep Suppl 10:vi+97
Delsarte P (1974) The association schemes of coding theory. In Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974), Part 1: Theory of designs, finite geometry and coding theory. Math. Centre Tracts, No. 55. Math. Centrum, Amsterdam, pp 139–157
P Frankl (1986) ArticleTitleOrthogonal vectors in the n-dimensional cube and codes with missing distances Combinatorica 6 279–285 Occurrence Handle0648.05031 Occurrence Handle875296
V Grolmusz (2000) ArticleTitleSuperpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs Combinatorica 20 73–88 Occurrence Handle1770535 Occurrence Handle10.1007/s004930070032
V Grolmusz (2002) ArticleTitleConstructing set-systems with prescribed intersection sizes J Algorithms 44 321–337 Occurrence Handle1014.68122 Occurrence Handle1932054 Occurrence Handle10.1016/S0196-6774(02)00204-3
V Grolmusz B Sudakov (2002) ArticleTitleOn k-wise set-intersections and k-wise Hamming-distances J Combinator Theory A 99 180–190 Occurrence Handle1013.05085 Occurrence Handle1911465 Occurrence Handle10.1006/jcta.2002.3264
Pless VS, Huffman WC, Brualdi RA (eds) (1998) Handbook of coding theory, vol. I, II. North-Holland, Amsterdam
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Communicated by S. Gao.
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Grolmusz, V. Pairs of codes with prescribed Hamming distances and coincidences. Des Codes Crypt 41, 87–99 (2006). https://doi.org/10.1007/s10623-006-0016-4
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DOI: https://doi.org/10.1007/s10623-006-0016-4