Codes defined by forms of degree 2 on hermitian surfaces and Sørensen's conjecture
Abstract
Description
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 $PG(3,q)$, Sørensen 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 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
accepted for publication in Finite Fields and Their Applications
accepted for publication in Finite Fields and Their Applications