Binary codes of the symplectic generalized quadrangle of even order

Article Type

Research Article

Publication Title

Designs, Codes, and Cryptography

Abstract

Let (Formula presented.) be a prime power and (Formula presented.) be the symplectic generalized quadrangle of order (Formula presented.). For (Formula presented.) even, let (Formula presented.) (respectively, (Formula presented.) , (Formula presented.) ) be the binary linear code spanned by the ovoids (respectively, elliptic ovoids, Tits ovoids) of (Formula presented.) and (Formula presented.) be the graph defined on the set of ovoids of (Formula presented.) in which two ovoids are adjacent if they intersect at one point. For (Formula presented.) , we describe the codewords of minimum and maximum weights in (Formula presented.) and its dual (Formula presented.) , and show that (Formula presented.) is a one-step completely orthogonalizable code (Theorem 1.1). We prove that, for (Formula presented.) , any blocking set of (Formula presented.) with respect to the hyperbolic lines of (Formula presented.) contains at least (Formula presented.) points and equality holds if and only if it is a hyperplane of (Formula presented.) (Theorem 1.3). We deduce that a clique in (Formula presented.) has size at most (Formula presented.) (Theorem 1.4).

First Page

163

Last Page

170

DOI

10.1007/s10623-015-0040-3

Publication Date

4-1-2016

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