Macaulay, Lazard and the Syndrome Variety

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