Symmetric bilinear forms over finite fields with applications to coding theory

Schmidt, Kai-Uwe

Journal of Algebraic Combinatorics, Tome 42 (2015) no. 2, pp. 635-670.

Résumé

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb {F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a commutative translation association scheme. We give explicit expressions for the eigenvalues of this scheme in terms of linear combinations of generalized Krawtchouk polynomials. We then study $d$-codes in this scheme, namely subsets $Y$ of $X(m,q)$ with the property that, for all distinct $A,B\in Y$, the rank of $A-B$ is at least $d$. We prove bounds on the size of a $d$-code and show that, under certain conditions, the inner distribution of a $d$-code is determined by its parameters. Constructions of $d$-codes are given, which are optimal among the $d$-codes that are subgroups of $X(m,q)$. Finally, with every subset $Y$ of $X(m,q)$, we associate two classical codes over $\mathbb {F}_q$ and show that their Hamming distance enumerators can be expressed in terms of the inner distribution of $Y$. As an example, we obtain the distance enumerators of certain cyclic codes, for which many special cases have been previously obtained using long ad hoc calculations.

Détail

Classification : 05E30, 15A63, 11T71, 94B15
Keywords: association scheme, symmetric bilinear form, quadratic form, code, weight enumerator