SYMMETRIC BILINEAR FORMS OVER FINITE FIELDS WITH APPLICATIONS TO CODING THEORY

Abstract. 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 Fq. 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 generalised 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 ∈ 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 Fq 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. 1