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Cyclic Projective Orbits on Rational Normal Curves and MDS Codes

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Pith's one-line read For r≥3 and n≥r+3, an MDS Krylov orbit segment lies on a rational normal curve exactly when the cyclic pair is conjugate to the (r−1)-st symmetric power of PGL2—over finite fields this gives a complete four-family classification of generali

desk verdict A solid, genuinely new rigidity theorem for cyclic MDS orbits on rational normal curves; referee it, with only small fixes needed. read the letter →

arxiv 2607.12761 v2 pith:GAL23TTN submitted 2026-07-14 math.NT

Yangcheng Li , Pingzhi Yuan This is my paper · ORCID
classification math.NT MSC 14N0594B0551E2111T71
keywords MDScodesrationalnormalcurvescyclicoperatorsKrylovgeneralizedReed–SolomoncompanionmatricesPGL2symmetricpowersalgebraictori
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies Krylov codes, whose parity-check matrix is a finite projective orbit segment (z, Az, …, A^{n−1}z) of a cyclic operator A. Its main theorem, for r≥3 and n≥r+3, is a rigidity statement: an MDS orbit segment lies on a rational normal curve precisely when the projective pair (A,[z]) is conjugate to the (r−1)-st symmetric-power action of PGL2. Over finite fields, applied to companion matrices, this yields a complete classification of the generalized Reed–Solomon (GRS) locus into split semisimple, two nonsplit semisimple, and unipotent families. The paper also shows the semisimple GRS coefficient closure is a two-dimensional rational surface, counts exactly how many GRS polynomials exist over F_q, and proves that, for fixed r and n, the proportion of monic polynomials whose companion codes are MDS but non-GRS tends to 1 as q grows.

What carries the argument

The central object is the cyclic pair (A,z) and its orbit segment [z],[Az],…,[A^{n−1}z], encoded as the kernel of the Krylov matrix H_n(A,z). Two facts carry the argument: the d+3-point uniqueness lemma—any d+3 points in linearly general position determine at most one rational normal curve—and the stabilizer computation showing that the projective stabilizer of the standard rational normal curve Γ0=ν_{r−1}(P^1) is exactly the image of the (r−1)-st symmetric-power representation of PGL2. The uniqueness lemma makes two overlapping blocks of r+2 orbit points force AΓ=Γ, and the stabilizer computation then identifies the projective class of A.

What would settle it

Exhibit a cyclic pair (A,z) over a field with r≥3 and n≥r+3 such that the Krylov code is MDS, the points [z],…,[A^{n−1}z] lie on a rational normal curve, yet A is not similar to c Sym^{r−1}(B) for any S, B, c. Equivalently, over F_q, find a monic polynomial g of degree r with an irreducible factor of degree at least 3 whose companion code is both MDS and GRS; the paper's Corollaries 4.6 and 5.5 predict no such g exists.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.4: if a cyclic pair (A,z) produces an MDS Krylov code and the projective points [z],[Az],…,[A^{n−1}z] lie on a k-form of a rational normal curve, then A is, up to a scalar, the (r−1)-st symmetric power of a 2×2 matrix B, and [z] is the image of a k-rational point of P^1 under the Veronese embedding. Conversely, any pair of that form has its orbit on the standard rational normal curve. The containing curve is unique, split, and preserved by the projective class of A. Over finite fields, Corollary 4.5 turns this into an absolute criterion: an MDS companion code is GRS exactly when its orbit arc is contained in a rational normal curve, and Theorem 5.3 lists the fo

Load-bearing premise

The load-bearing premise is the classical d+3-point uniqueness lemma: over an algebraic closure, any d+3 points in linearly general position lie on at most one rational normal curve; the entire rigidity argument—that two overlapping blocks of orbit points force A to preserve the curve—depends on this uniqueness, and the stable range n≥r+3 is exactly what guarantees the required number of general-position points.

Editorial extensions

If this is right

  • An MDS Krylov code is of GRS type exactly when its projective orbit arc is contained in a rational normal curve over the base field, after which the containing curve is necessarily split.
  • Over finite fields, the GRS locus among companion codes splits into four mutually exclusive arithmetic families: unipotent pure powers, split semisimple geometric progressions, and two nonsplit semisimple families distinguished by parity of r.
  • Any MDS companion code whose polynomial has an irreducible factor of degree at least 3, or which is non-squarefree without being a pure power, is automatically non-GRS.
  • The exact number of GRS polynomials over F_q is (q−1)·1_{n≤p} + (q−1)/2 (E_{q−1}(n)+E_{q+1}(n)), where E_m(n) counts elements of order at least n in a cyclic group of order m.
  • For fixed r≥3 and n≥r+3, the proportion of monic degree-r polynomials whose companion codes are MDS but non-GRS tends to 1 as q goes to infinity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The codimension r−2 of the GRS coefficient surface explains why no single low-degree polynomial can recognize GRS companion codes for r≥4; recognition must involve higher-degree or multi-component conditions.
  • The generic two-to-one parameterization of the coefficient surface by reversal suggests an algorithmic route for GRS testing: decide whether a companion polynomial arises as a geometric progression with ratio of order at least n, which is a finite torus-quotient problem.
  • A natural refinement of the counting theorem is to count monomial-equivalence classes of GRS codes rather than generator polynomials, since the paper's formula counts polynomials and the authors explicitly separate the two questions.
  • The stability threshold n≥r+3 is sharp in an interesting way: Appendix A shows that for n=r+1 or n=r+2 every MDS companion code is GRS, so the non-GRS density phenomenon only appears once the orbit segment is long enough to engage the rigidity mechanism.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper defines Krylov codes for cyclic operator-vector pairs (A,z) via the kernel of the Krylov matrix (z,Az,...,A^{n-1}z) and studies when such an MDS orbit segment lies on a rational normal curve. The main rigidity theorem (Theorem 4.4) asserts that for r≥3 and n≥r+3, an MDS Krylov orbit lies on an RNC iff the projective pair is conjugate to the symmetric-power action of PGL_2; the containing RNC is then unique, split, and A-stable. For companion operators over finite fields, Theorem 5.3 gives a complete classification of the GRS locus into unipotent (U), split semisimple (S), and two nonsplit semisimple families (N0,N1). The second half of the paper studies the coefficient-space geometry of the semisimple GRS locus: it is a two-dimensional rational surface, its normalization over the nonzero-constant-term part is the affine quotient of a two-dimensional torus by the reversal involution, Frobenius descent yields an exact finite-field counting formula (Theorem 6.14), and the MDS non-GRS proportion tends to one as q→∞ (Theorem 6.15). Boundary lengths n=r+1,r+2 are treated in Appendix A.

Significance. If correct, this is a substantial contribution: it gives a clean geometric and representation-theoretic characterization of when cyclic orbit arcs are GRS, a complete finite-field companion classification, and an exact arithmetic count. The paper is largely self-contained and careful about characteristic issues; the proof of the main rigidity theorem rests on standard Castelnuovo uniqueness and the RNC stabilizer, both handled explicitly. Concrete reproducible examples are provided in Section 7. The authors are also appropriately explicit about the scope of the normalization theorem and about the boundary ranges. I found no load-bearing technical objection; the issues I identified are local and easily fixed.

minor comments (4)
  1. [§6.5, proof of Theorem 6.13] The first line of the proof says 'By Theorem 6.2(5), applied with k = F_q'. Theorem 6.2(5) gives the surjectivity on k-points only for algebraically closed k. The proof should apply Theorem 6.2(5) with k = \overline{F_q} and then use the Frobenius dichotomy that follows. The subsequent counting theorem (Theorem 6.14) is proved directly from Theorem 5.3, so this is a local typographical-level flaw, not a substantive gap.
  2. [Appendix B, proof of Proposition B.1] The proof says to choose d+2 pairwise distinct elements u_0,...,u_{d+1} in k. This is impossible when |k|<d+2, e.g. k=F_4 and r=3. Since [N] already fixes every geometric point of Γ0, the frame should be chosen over \bar{k}; then a projectivity fixing d+2 points in linearly general position over \bar{k} is the identity, so the conclusion follows. The statement is true, but the proof needs this small correction to cover small finite fields.
  3. [§4.4, proof of Corollary 4.5] The ideal I(Γ) should be in \overline{F_q}[X_0,...,X_{r-1}], not F_q[X_0,...,X_{r-1}]; as printed, the Frobenius action and Galois descent step do not make sense. This is a notation typo and does not affect the argument once corrected.
  4. [§4.3, proof of Theorem 4.4] When invoking Lemma 4.3, the text says 'applied over k', but the lemma is stated for algebraically closed fields. This should be read as applying after base change to \bar{k}; the accompanying uniqueness argument then goes through. A one-sentence clarification would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the orbit-rigidity and classification results are derived from external RNC uniqueness/stabilizer facts and PGL2 conjugacy, not from their own conclusions.

full rationale

The paper's central orbit-rigidity statement (Theorem 4.4) is not circular. The forward direction combines the d+3-point Castelnuovo uniqueness lemma (cited from Harris [9, Thm 1.18]) with the standard computation of the RNC stabilizer (proved in Appendix B as Proposition B.1). The cited uniqueness lemma is an external, characteristic-free benchmark, not a result of the present authors, and the proof verifies that the relevant d+3 points are in linearly general position via the MDS hypothesis before applying it. The reverse direction is a direct equivariance argument. The finite-field classification (Theorem 5.3) follows from Theorem 4.4 plus the three PGL2(F_q) conjugacy types; the sufficiency constructions verify cyclicity and MDS by explicit Vandermonde/Pascal determinant arguments. The coefficient-surface theorems (6.2, 6.3) use torus quotients, invariant rings, and normalization; they do not assume the target classification. The exact count in Theorem 6.14 is a direct count of parameter pairs modulo the reversal involution, with no fitted parameters. The asymptotic genericity bound (Theorem 6.15) uses only the nonzero reduction of D_{n,r} (Proposition 3.6) and a standard polynomial zero bound (Lemma 3.7). There are no load-bearing self-citations. The only manuscript issue I noticed is typographical: in the proof of Theorem 6.13, Theorem 6.2(5) should be applied with k = \overline{F_q} before the Frobenius descent step, not with k = F_q; this affects wording only and does not alter the theorem or the counting formula. No circular step was found.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted constants appear: r, n, q, and k are theorem inputs, while γ, t, β are coordinates on parameter tori rather than ad hoc fitted values. No new physical or mathematical entity is postulated; G_{r,k}, the reversal involution, and the quotient T/⟨ι⟩ are constructed mathematical objects, not entities requiring independent falsifiable evidence.

assumptions (7)
  • standard math Castelnuovo d+3-point uniqueness for rational normal curves (Lemma 4.3; [9, Thm 1.18]).
    Used in Thm 4.4 to conclude AΓ=Γ from two overlapping RNCs; accepted without proof.
  • standard math Projective stabilizer of the standard RNC equals the image of Sym^{r-1}: PGL2 -> PGL_r (Prop B.1).
    Proved in Appendix B; this identification makes symmetric-power membership equivalent to RNC preservation.
  • standard math Faithfully flat / Galois descent for subschemes and graded ideals over finite fields (Cor 4.5).
    Used to descend a geometric RNC to an F_q-form; invoked without proof.
  • standard math Finite-group quotient facts: primes over an invariant prime form one orbit; free invariant open is a torsor (Cor 6.4; Stacks Tags 0BRI, 07S7).
    Used to identify fixed loci and obtain the finite-étale degree-two quotient.
  • standard math PGL2(F_q) conjugacy types: split semisimple, nonsplit semisimple, unipotent (Section 5).
    Basis of the four-family finite-field classification.
  • standard math Lucas theorem and modular binomial behavior in p-characteristic (Lemma 5.2).
    Controls vanishing binomial coefficients and sparse multiples for pure powers.
  • standard math Surjectivity of x↦x^{q-1}: F_{q^2}^* → norm-one subgroup, with kernel F_q^* (Thm 5.3, Thm 6.14).
    Used to construct λ and to count γ solutions in the nonsplit semisimple family.

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Pith. "Pith review of Cyclic Projective Orbits on Rational Normal Curves and MDS Codes." pith.science (2026). https://pith.science/paper/GAL23TTN

@misc{pith2026260712761,
  author       = {Pith},
  title        = {Pith review of: Cyclic Projective Orbits on Rational Normal Curves and MDS Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GAL23TTN}},
  note         = {Machine review of arXiv:2607.12761}
}
abstract

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.

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

    cs.IT 2026-08 accept novelty 7.0 of 10

    Every normalized maximal Plücker coordinate of the companion remainder orbit equals, up to sign and a power of the constant coefficient, a Schur polynomial, so the universal MDS polynomial is A0 times all Schur polyno...

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