Codes defined by forms of degree 2 on hermitian surfaces and Sørensen's conjecture

2006/12/09 by Frédéric A. B. Edoukou, Frederic A. B. Edoukou, Edoukou, Frederic A. B.
Computer Science · Engineering · Mathematics · #05B25 #11T71 #14J29 #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems #math.AG #msc:05B25 #msc:11T71 #msc:14J29

paper · pdf · doi:10.48550/arxiv.math/0612231

accepted for publication in Finite Fields and Their Applications

arxiv created 2006/12/09 · openalex publication_date 2006/12/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the functional codes C_h(X) defined by G. Lachaud in \lbrack 10 \rbrack where X ⊂ ℙ^N is an algebraic projective variety of degree d and dimension m. When X is a hermitian surface in PG(3,q), Sørensen in \lbrack 15\rbrack, has conjectured for h≤ t (where q=t²) the following result : # X_Z(f)(\mathbbF_q) ≤ h(t³+ t²-t)+t+1 which should give the exact value of the minimum distance of the functional code C_h(X). In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. h=2), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight

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