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

Details Codes defined by forms of degree 2 on hermitian surfaces and Sø rensen's conjecture Codes defined by forms of degree 2 on hermitian surfaces and Sø rensen's conjecture

2006-12-09

arxiv.org/abs/math/0612231v1

Mathematics

Algebraic Geometry

accepted for publication in Finite Fields and Their Applications

Scientific paper

We study the functional codes $C_h(X)$ defined by G. Lachaud in $\lbrack 10 \rbrack$ where $X \subset {\mathbb{P}}^N$ is an algebraic projective variety of degree $d$ and dimension $m$. When $X$ is a hermitian surface in $PG(3,q)$, S{\o}rensen in \lbrack 15\rbrack, has conjectured for $h\le t$ (where $q=t^2$) the following result : $$# X_{Z(f)}(\mathbb{F}_{q}) \le h(t^{3}+ t^{2}-t)+t+1$$ 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 S{\o}rensen 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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