Cyclic Projective Orbits on Rational Normal Curves and MDS Codes

Let $A$ be a cyclic operator on an $r$-dimensional vector space over a field $k$, and let $z$ be a cyclic vector. Their Krylov code has parity-check matrix $(z,Az,\dots ,A^{n-1}z)$. For $r\ge 3$ and $n\ge r+3$, we prove that an MDS orbit segment lies on a rational normal curve precisely when the projective pair $(A,[z])$ is conjugate to one arising from the $(r-1)$-st symmetric-power action of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$. Over finite fields, for companion operators, this gives a complete classification of the generalized Reed–Solomon locus into split semisimple, two nonsplit semisimple, and unipotent families. Over an algebraically closed field $k$, the Zariski closure ${\text{GRSsurf}}_{r,k}$ of the semisimple GRS coefficient locus is an irreducible rational surface, generically parameterized two-to-one by a two-dimensional torus of geometric-progression root sets; reversal is the generic ambiguity. The affine quotient of the parameter torus by reversal is the normalization of ${\text{GRSsurf}}_{r,k}\cap D(a_0)$, its nonzero-constant-term open part. The codimension in the space of monic degree-$r$ polynomials is $r-2$. Frobenius descent gives an exact formula for the number of GRS polynomials over ${\mathbb{F}}_q$. A canonical remainder parity-check matrix defines the MDS locus by a principal open condition. For fixed $r\ge 3$ and $n\ge r+3$, the proportion of all monic degree-$r$ polynomials over ${\mathbb{F}}_q$ whose companion codes are MDS and non-GRS tends to one as $q\to \mathrm{\infty }$ through prime powers.