Symmetric bilinear forms over finite fields with applications to coding theory

Abstract

Let $$q$$q be an odd prime power and let $$X(m,q)$$X(m,q) be the set of symmetric bilinear forms on an $$m$$m-dimensional vector space over $$\mathbb {F}_q$$Fq. The partition of $$X(m,q)$$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 $$d$$d-codes in this scheme, namely subsets $$Y$$Y of $$X(m,q)$$X(m,q) with the property that, for all distinct $$A,B\in Y$$A,B?Y, the rank of $$A-B$$A-B is at least $$d$$d. We prove bounds on the size of a $$d$$d-code and show that, under certain conditions, the inner distribution of a $$d$$d-code is determined by its parameters. Constructions of $$d$$d-codes are given, which are optimal among the $$d$$d-codes that are subgroups of $$X(m,q)$$X(m,q). Finally, with every subset $$Y$$Y of $$X(m,q)$$X(m,q), we associate two classical codes over $$\mathbb {F}_q$$Fq and show that their Hamming distance enumerators can be expressed in terms of the inner distribution of $$Y$$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.