2018/03/12 by Kai‐Uwe Schmidt, Schmidt, Kai-Uwe · 1 citation
Computer Science · Social Sciences · #15A63 #94B15 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Islamic Finance and Communication #Primary 05E30 #Secondary 11T71
paper · pdf · doi:10.48550/arxiv.1803.04274
openalex publication_date 2018/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let mathscrQ(m,q) and mathscrS(m,q) be the sets of quadratic forms\nand symmetric bilinear forms on an m-dimensional vector space over\n mathbbFq, respectively. The orbits of mathscrQ(m,q) and\n mathscrS(m,q) under a natural group action induce two translation\nassociation schemes, which are known to be dual to each other. We give explicit\nexpressions for the eigenvalues of these association schemes in terms of linear\ncombinations of generalised Krawtchouk polynomials, generalising earlier\nresults for odd q to the more difficult case when q is even. We then study\nd-codes in these schemes, namely subsets X of mathscrQ(m,q) or\n mathscrS(m,q) with the property that, for all distinct A,B\∈ X, the\nrank of A-B is at least d. We prove tight bounds on the size of d-codes\nand show that, when these bounds hold with equality, the inner distributions of\nthe subsets are often uniquely determined by their parameters. We also discuss\nconnections to classical error-correcting codes and show how the Hamming\ndistance distribution of large classes of codes over mathbbFq can be\ndetermined from the results of this paper.\n