2014/10/27 by Kai‐Uwe Schmidt, Schmidt, Kai-Uwe · 1 citation
Computer Science · Mathematics · Social Sciences · #05E30 #94B15 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #Islamic Finance and Communication #Primary: 15A63 #Secondary: 11T71
paper · pdf · doi:10.48550/arxiv.1410.7184
openalex publication_date 2014/10/27 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
Let q be an odd prime power and let X(m,q) be the set of symmetric\nbilinear forms on an m-dimensional vector space over mathbbFq. The\npartition of X(m,q) induced by the action of the general linear group gives\nrise to a commutative translation association scheme. We give explicit\nexpressions for the eigenvalues of this scheme in terms of linear combinations\nof generalised Krawtchouk polynomials. We then study d-codes in this scheme,\nnamely subsets Y of X(m,q) with the property that, for all distinct A,B\∈\nY, the rank of A-B is at least d. We prove bounds on the size of a\nd-code and show that, under certain conditions, the inner distribution of a\nd-code is determined by its parameters. Constructions of d-codes are given,\nwhich are optimal among the d-codes that are subgroups of X(m,q). Finally,\nwith every subset Y of X(m,q), we associate two classical codes over\n mathbbFq and show that their Hamming distance enumerators can be\nexpressed in terms of the inner distribution of Y. As an example, we obtain\nthe distance enumerators of certain cyclic codes, for which many special cases\nhave been previously obtained using long ad hoc calculations.\n