Symmetric bilinear forms over finite fields with applications to coding theory
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- 作者：Kai-Uwe Schmidt
- 关键词：Association scheme ; Symmetric bilinear form ; Quadratic form ; Code ; Weight enumerator ; Primary 05E30 ; 15A63 ; Secondary 11T71 ; 94B15
- 刊名：Journal of Algebraic Combinatorics
- 出版年：2015
- 出版时间：September 2015
- 年：2015
- 卷：42
- 期：2
- 页码：635-670
- 全文大小：655 KB
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1. Faculty of Mathematics, Otto-von-Guericke University, Universit?tsplatz?2, 39106, Magdeburg, Germany - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Combinatorics
Convex and Discrete Geometry
Order, Lattices and Ordered Algebraic Structures
Computer Science, general
Group Theory and Generalizations - 出版者：Springer U.S.
- ISSN：1572-9192
文摘
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 generalized 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.