Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes

Abstract

Let \(g\) be a monic polynomial of degree \(r<n\), and let \(C_g(n)\) be the coefficient-vector code formed by multiples \(ug\) with \(\deg(ug)<n\). We study the coefficient-space MDS locus \(M_{n,r}\). The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of \(\operatorname{Gr}(r,n)\). We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient \(A_0\) times a Schur polynomial \(S_\kappa(g)=s_\kappa(\Lambda_g)\), where \(\kappa\subseteq (n-r)^{r-1}\). Hence the universal MDS polynomial is \[D_{n,r}=A_0\prod_{\kappa\subseteq (n-r)^{r-1}}S_\kappa.\] This description yields a flat non-MDS boundary over \(\mathbb{Z}\), explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for \(r\ge 3\) and \(N\ge r+3\), every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.