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.
Citation: Lijun Ma, Shuxia Liu, Zihong Tian. The binary codes generated from quadrics in projective spaces[J]. AIMS Mathematics, 2024, 9(10): 29333-29345. doi: 10.3934/math.20241421
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.
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Lijun Ma, Shuxia Liu, Zihong Tian. The binary codes generated from quadrics in projective spaces[J]. AIMS Mathematics, 2024, 9(10): 29333-29345. doi: 10.3934/math.20241421
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