Quadratic and symmetric bilinear forms over finite fields and their association schemes

Abstract: Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a natural group action induce two translation association schemes, which are known to be dual to each other. We give explicit expressions for the eigenvalues of these association schemes in terms of linear combinations of generalised Krawtchouk polynomials, generalising earlier results for odd $q$ to the more difficult case when $q$ is even. We then study $d$-codes in these schemes, namely subsets $X$ of $\mathscr{Q}(m,q)$ or $\mathscr{S}(m,q)$ with the property that, for all distinct $A,B\in X$, the rank of $A-B$ is at least $d$. We prove tight bounds on the size of $d$-codes and show that, when these bounds hold with equality, the inner distributions of the subsets are often uniquely determined by their parameters. We also discuss connections to classical error-correcting codes and show how the Hamming distance distribution of large classes of codes over $\mathbb{F}_q$ can be determined from the results of this paper.