2026/08/02 by Yangcheng Li, Pingzhi Yuan
Computer Science · Mathematics · #cs.IT #math.IT #math.NT
48 pages
arxiv created 2026/08/02 · arxiv updated 2026/08/04
Let \(g\) be a monic polynomial of degree \(r<n\), and let \(Cg(n)\) be the coefficient-vector code formed by multiples \(ug\) with \(°(ug)<n\). We study the coefficient-space MDS locus \(Mn,r\). 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 \(Gr(r,n)\). We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient \(A0\) times a Schur polynomial \(Sκ(g)=sκ(Λg)\), where \(κ⊆ (n-r)r-1\). Hence the universal MDS polynomial is Dn,r=A0∏κ⊆ (n-r)r-1Sκ. This description yields a flat non-MDS boundary over \(ℤ\), 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 \(r≥ 3\) and \(N≥ r+3\), every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.