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Symmetric bilinear forms over finite fields with applications to coding theory
- Publication Year :
- 2014
-
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.<br />Comment: 33 pages
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1410.7184
- Document Type :
- Working Paper