The binary codes generated from quadrics in projective spaces

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^{\perp}(M) $ in PG$ (2, q) $ and give their upper bounds.

Keywords

• quadric
• conic
• binary code
• generator matrix
• dimension