Macaulay, Lazard and the Syndrome Variety

Abstract

In this paper we consider the four syndrom varieties ${\mathsf{Z}}_e^{\times }$, i.e. the set of all error locations corresponding to errors of weight $w,0\le w\le 2$, ${\mathsf{Z}}_{ns}^{\times }$ , the set of all {\em non spurious} error locations corresponding to errors of weight $w,0\le w\le 2$, ${\mathsf{Z}}_{+}^{\times }$, the set of all non-spurious error locations corresponding to errors of weight $w,1\le w\le 2$, ${\mathsf{Z}}_2^{\times }$, the set of all non-spurious error locations corresponding to errors of weight $w=2$, associated to an up-to-two errors correcting binary cyclic codes. Denoting $J_{\ast }:=\mathcal{I}({\mathsf{Z}}_{\ast })$, the ideal of these syndrome varieties, ${\mathsf{N}}_{\ast }:=\mathbf{N}(J_{\ast })$ the \GR\ escalier of $J_{\ast }$ w.r.t. the lex ordering with $x_1<x_2<z_1<z_2$, ${\Phi }_{\ast }:{\mathsf{Z}}_{\ast }\to {\mathsf{N}}_{\ast }$ a Cerlienco-Mureddu correspondence, and $G_{\ast }$ a minimal Groebner basis of the ideal $J_{\ast }$, the aim of the paper is, assuming to know the structure of the order ideal ${\mathsf{N}}_2$ and a Cerlienco Mureddu Correspondence to deduce with elementary arguments ${\mathsf{N}}_{\ast }$, $G_{\ast }$ and ${\Phi }_{\ast }$ for $\ast \in \{e,ns,+\}$. The tools are Macaulay's trick and Lazard's formulation of Cerlienco-Mureddu correspondence.

Very preliminary version