The binary codes generated from quadrics in projective spaces
Lijun Ma, Shuxia Liu, Zihong Tian · AIMS Mathematics · 2024
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.