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

Kai-Uwe Schmidt

doi:10.5802/alco.88

Abstract

Let 𝒬(m,q) and 𝒮(m,q) be the sets of quadratic forms and symmetric bilinear forms on an m-dimensional vector space over 𝔽 q , respectively. The orbits of 𝒬(m,q) and 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 𝒬(m,q) or 𝒮(m,q) with the property that, for all distinct A,B∈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 𝔽 q can be determined from the results of this paper.

Highlights

Let q be a prime power and let V = V (m, q) be an m-dimensional Fq-vector space
The main motivation for this paper is to study d-codes in Q and S, namely subsets X of Q or S such that, for all distinct A, B ∈ X, the rank of A − B is at least d
One of the applications is that d-codes in Q can be used to construct optimal subcodes of the second-order generalised Reed–Muller code and our theory can be used to determine the Hamming distance distributions of such codes

Summary

Introduction

The main result of the present paper is the determination of the P - and Q-numbers of Q and, by duality, the Q- and P -numbers of S These numbers do not directly arise from evaluations of orthogonal polynomials, they can be expressed in terms of linear combinations of generalised Krawtchouk polynomials. We show that the Hamming distance enumerators of these error-correcting codes are uniquely determined by their parameters This at once gives the distance enumerators of large classes of errorcorrecting codes for which many special cases have been obtained previously using different methods, for example results for (extended) binary cyclic codes obtained by Berlekamp [4] and Kasami [14], recent results for q-ary cyclic codes obtained by Li [15], and many results for q-ary cyclic codes and odd q, as explained in [21]. These results are recovered in this paper and the distance enumerators are obtained for all q as corollaries of our results

Quadratic forms and symmetric bilinear forms

Computation of the Q- and P -numbers

E UT UB and

Subsets of quadratic and symmetric bilinear forms

Constructions

Applications to classical coding theory