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Research Article | Open Access

The binary codes generated from quadrics in projective spaces

Lijun Ma, Shuxia Liu, Zihong Tian( )
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China
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Abstract

Quadrics are important in finite geometry and can be used to construct binary codes. In this paper, we first define an incidence matrix M based on points and non-degenerate quadrics in the classical projective space PG (n−1,q), where q is a prime power. As a consequence, we establish a binary code C(M) with the generator matrix M and determine the dimension of C(M) when q and n are both odd. In particular, we study the minimum distances of C(M) and C⊥(M) in PG (2,q) and give their upper bounds.

Keywords

quadricconicbinary codegenerator matrixdimension
CLC number: 05B25, 15A03, 51E20, 94B05

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AIMS Mathematics
Volume 9 Issue 10,
October 2024
Pages 29333-29345
DOI: 10.3934/math.20241421

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Cite this article:
Ma L, Liu S, Tian Z. The binary codes generated from quadrics in projective spaces. AIMS Mathematics, 2024, 9(10): 29333-29345. https://doi.org/10.3934/math.20241421

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Received: 09 August 2024
Revised: 16 September 2024
Accepted: 29 September 2024
Published: 15 October 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)