Symmetric bilinear forms over finite fields with applications to coding theory

TL;DR

This paper studies symmetric bilinear forms over finite fields, develops eigenvalue formulas for an associated scheme, and explores bounds and constructions of special subsets called d-codes with applications to classical coding theory.

Contribution

It introduces explicit eigenvalue expressions for a symmetric bilinear form scheme and provides bounds, constructions, and applications of d-codes in coding theory.

Findings

Eigenvalues expressed via generalized Krawtchouk polynomials.

Bounds on the size of d-codes established.

Connections between subsets of forms and classical codes demonstrated.

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 $\mathbb{F}_q$. 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 $A,B\in Y$, 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.