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 \((z,Az,\ldots,A^{n-1}z)\). 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 \(\mathrm{PGL}_2\). 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 \(\GRSsurf_{r,k}\) 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 \(\GRSsurf_{r,k}\cap D(a_0)\), 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 \(r\ge3\) 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 \(q\to\infty\) through prime powers.