Symmetric bilinear forms over finite fields with applications to coding theory

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 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 $A,B\in Y$,therankof$A-B$isatleast$d$\mathrm{.}Weproveboundsonthesizeofa$d$-codeandshowthat,undercertainconditions,theinnerdistributionofa$d$-codeisdeterminedbyitsparameters\mathrm{.}Constructionsof$d$-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.