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

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Pith's one-line read This paper proves that every maximal minor of the coefficient-vector remainder matrix for a monic degree-$r$ polynomial $g$ factors, up to a power of $a_0$, as a Schur polynomial evaluated at the roots of $g$; consequently the MDS test…

desk verdict The Schur–Plücker factorization is real and proven in-paper; the non-GRS density claims are real only modulo the companion preprint [12], which is the right thing to flag in review. read the letter →

arxiv 2608.01146 v1 pith:VVJYCHT4 submitted 2026-08-02 cs.IT math.ITmath.NT

Yangcheng Li , Pingzhi Yuan This is my paper · ORCID
classification cs.ITmath.ITmath.NT MSC 94B0505E0511T7114M15
keywords MDScodesprincipal-idealSchurpolynomialsPlückercoordinatesGrassmanniansconfluentVandermondedeterminantssparsemultipleslengthfiltration
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 coefficient-vector codes $C_g(n)$ formed by all multiples of a monic degree-$r$ polynomial $g$ with degree below $n$, and asks exactly when such a code is MDS, meaning any $r$ positions determine the whole codeword. Using the remainder orbit of $g$ inside the Grassmannian $\operatorname{Gr}(r,n)$, it proves that every maximal minor (normalized Plücker coordinate) is, up to a power of the constant coefficient $a_0$, a Schur polynomial $S_\kappa$ evaluated at the roots of $g$. Hence the universal MDS polynomial is the single product $D_{n,r}=a_0\prod_{\kappa\subseteq (n-r)^{r-1}}S_\kappa$: the code is MDS if and only if $a_0$ and every $S_\kappa(g)$ are nonzero. This turns the MDS locus into the complement of a flat boundary divisor with explicit degree, yields finite-field counts, and identifies the first length where MDS fails with the first sparse multiple of $g$. It also shows that, on nonempty first-failure layers, MDS members outside the generalized Reed–Solomon family are dense.

What carries the argument

The machinery is the remainder orbit embedded in the standard big cell of $\operatorname{Gr}(r,n)$: the map $\Phi_{n,r}(g)=\operatorname{rowspan}H_g^{\mathrm{rem}}$ is a closed immersion whose image is the graph of the companion recurrence, cut out by a regular sequence of $r(n-r-1)$ equations. The master identity reduces each Plücker coordinate $p_I(\Phi_{n,r}(g))$ to $\det(T_g)^{i_0}\delta_{U_I}(g)$, where $T_g$ is the companion multiplication matrix and $\det(T_g)=(-1)^r a_0$. The bijection $U\mapsto\kappa(U)$ from reduced index sets to partitions in $(n-r)^{r-1}$, together with the bialternant formula, realizes every reduced minor $\delta_U(g)$ as the Schur polynomial $s_{\kappa(U)}$ evaluated at the root multiset of $g$; a confluent Hasse-derivative Vandermonde identity extends this to repeated roots without factorial factors. The Schur coordinates are then packaged into the single rectangular product $D_{n,r}=a_0\prod_{\kappa}S_\kappa$.

What would settle it

For a concrete counterexample, take $r=3$ and $n=6$ over several small finite fields, compute all $20$ maximal minors of $H_g^{\mathrm{rem}}=(Q_0\ \cdots\ Q_5)$, and compare their vanishing pattern with the ten-factor Schur product displayed in Example 7.3; any specialization where the product and the minors disagree would refute the master theorem. The identity is characteristic-free, so the same check can be run over any prime field and any extension.

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Extended reading notes

Core claim

The central discovery is an exact Schur–Plücker factorization for the coefficient-space MDS locus. For the remainder parity-check matrix $H_g^{\mathrm{rem}}=(Q_0\ \cdots\ Q_{n-1})$, where $Q_i$ is the coefficient vector of $x^i\bmod g$, the maximal minor indexed by $I=\{i_0<\cdots<i_{r-1}\}$ equals $\det(Q_{i_0},\dots,Q_{i_{r-1}})=((-1)^r a_0)^{i_0} S_{\kappa(U_I)}(g)$, with $\kappa(U_I)$ ranging over exactly the partitions in the rectangle $(n-r)^{r-1}$. Therefore the universal MDS polynomial satisfies $D_{n,r}=a_0\prod_{\kappa\subseteq(n-r)^{r-1}}S_\kappa$, and over every field $C_g(n)$ is MDS if and only if $a_0\neq 0$ and every $S_\kappa(g)\neq 0$. This one rectangular Schur system controls the flat non-MDS boundary over $\mathbb{Z}$, the explicit total degree, finite-field lower bounds, a length filtration with flat outer Cartier layers, and the exact sparse-multiple threshold $\sigma_r(g)=r+\tau(g)-1$. 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.

Load-bearing premise

The non-GRS classification and counting claims inherit the correctness and field-generality of the companion paper's rational-normal-curve rigidity, pure-power criterion, and exact GRS count; if any of those companion results fails over some field, the non-GRS density and counting statements would need revision, though the Schur–Plücker core would survive.

Editorial extensions

If this is right

  • The MDS property for every code $C_g(n)$ is decided by nonvanishing of the explicit polynomial $D_{n,r}=a_0\prod_{\kappa\subseteq(n-r)^{r-1}}S_\kappa$, so testing membership in the coefficient-space MDS locus is one algebraic test rather than a collection of unrelated minor equations.
  • The non-MDS boundary is a flat effective Cartier divisor over $\mathbb{Z}$ of exact total degree $\delta_{n,r}=1+\frac{(r-1)(n-r)}{r}\binom{n-1}{r-1}$, and over $\mathbb{F}_q$ the MDS locus has at least $q^r-\delta_{n,r}q^{r-1}$ points; for fixed $r$ with $n(q)^r=o(q)$, the MDS density tends to $1$.
  • For each $g$ with $a_0\neq 0$, the first length $N$ at which $C_g(N)$ stops being MDS equals $\sigma_r(g)=r+\tau(g)-1$, where $\tau(g)$ is the first vanishing Schur width, and this is also the degree of the first nonzero multiple of $g$ with at most $r$ nonzero coefficients.
  • For $r\ge 3$ and $N\ge r+3$, over an algebraically closed field the GRS-type coefficient points in every nonempty first-failure layer $X_{N,r}$ lie in a set of dimension at most one (codimension at least $r-2$), so the non-GRS locus is dense in every irreducible component of the layer.
  • The first-failure subschemes $X_{N,r}$ are flat local complete intersections over $\mathbb{Z}[1/N]$, giving a geometrically strict decreasing filtration of MDS open sets over fields of characteristic zero or prime to $N$.

Reading between the lines

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

  • Because the MDS test is a complete rectangle of Schur polynomials, the geometry of this code family is governed by symmetric-function combinatorics rather than by arbitrary polynomial systems; one could try to carry the same rectangular Schur system to other coefficient spaces defined by quotient algebras.
  • The identity $\sigma_r(g)=r+\tau(g)-1$ suggests a direct route to the sparse-multiple degree: evaluate rectangular Schur coordinates instead of searching for sparse polynomial multiples, which is computationally hard for general inputs.
  • The bad-characteristic bound $p<n$ on root-multiplicity strata, together with the flat outer layers, points toward a design rule over small-characteristic fields: repeated-root principal-ideal codes fail MDS at predictably short lengths, so the Schur coordinates can certify when such codes are safe to use.
  • A natural testable extension is to convert the density result into explicit non-GRS MDS families by prescribing Schur evaluations; the worked examples in the paper show the construction is feasible in small cases and likely algorithmic in general.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies coefficient-vector codes C_g(n)={u g : deg u < n-r} for a monic degree-r polynomial g, and the coefficient-space MDS locus M_{n,r}. It embeds coefficient space into the big cell of Gr(r,n) via the remainder-orbit columns and proves the master identity that every normalized maximal Plücker coordinate pulls back to ((-1)^r a_0)^{i0} S_{\kappa(U)}(g), where S_\kappa is a Schur polynomial in the root multiset of g. Hence the universal MDS polynomial is D_{n,r}=A_0 \prod_{\kappa\subseteq (n-r)^{r-1}} S_\kappa, and C_g(n) is MDS exactly when a_0\neq 0 and all S_\kappa(g)\neq 0. From this description the paper derives flatness of the universal boundary over Z, exact degree formulas, finite-field estimates, dilation and reciprocity symmetries, a length filtration with first-failure layers, multiplicity-stratum bad-characteristic criteria, density results on the irreducible Frobenius stratum, and the sparse-threshold identity \sigma_r(g)=r+\tau(g)-1. It then invokes the companion classification [12] to claim that, for r\geq 3 and N\geq r+3, every nonempty first-failure layer has a dense open non-GRS locus.

Significance. The core Theorem 3.3 is a clean, characteristic-free structural result: it identifies all MDS minors of this principal-ideal family with one complete rectangle of Schur polynomials. This is a substantial advance over treating the MDS test as an unrelated collection of minor equations, and it yields explicit and falsifiable consequences: the universal MDS polynomial, flat boundary divisors, exact degree formulas, finite-field lower bounds, and the exact sparse-threshold relation. The paper is also transparent about where it uses the companion preprint [12]; those external results are confined to the GRS-type applications and the quantitative non-GRS statements. If the companion theorems are correct, the paper makes a strong contribution to both coding theory and the algebraic geometry of the MDS locus.

major comments (2)
  1. [Abstract, Corollary 6.19, Eq. (6.39)] The advertised claim that, for r≥3 and N≥r+3, every nonempty first-failure layer has a dense open non-GRS locus is not self-contained in this manuscript. The proof of Corollary 6.19 invokes [12, Theorem 6.6] to classify GRS points and, through Proposition 4.11, [12, Lemma 4.3]; Corollary 4.10 uses [12, Theorem 6.14]; and Theorem 5.5 uses [12, Lemma 5.2]. Since [12] is a companion preprint whose statements are not reproduced, the final sentence of the abstract and the stated non-GRS density claims are conditional on external results. Please state this conditionality explicitly in the abstract and in each GRS-type corollary, or include the relevant companion statements with proofs in an appendix. This does not affect Theorem 3.3, but it is load-bearing for the paper's last advertised contribution.
  2. [Corollary 4.10, Propositions 5.13 and 5.14] The quantitative non-GRS results subtract or invoke exact counts and classifications imported from [12]. In particular, Corollary 4.10 (Eqs. (4.20)–(4.21)) subtracts #G_{n,r}(F_q)=[12, Theorem 6.14] exactly, and Propositions 5.13 and 5.14 rely on the factorwise classification [12, Theorem 5.3] without proof. If any of those companion theorems fails over fields of small characteristic or over non-finite fields, the corresponding lower bounds and factor criteria in this paper would need revision. The dependence is acknowledged in Remarks 4.12 and 6.16, but acknowledgment does not make the result self-contained; the statements should be labeled as conditional on [12] or the missing companion proofs should be supplied.
minor comments (3)
  1. [§4, Eq. (4.27)] The determinant identity in Proposition 4.11 is stated after 'direct substitution'; I checked it in a generic case and it is correct. A one-line intermediate expansion would improve verifiability, but this is not a correctness issue.
  2. [Example 7.5, length-seven factors] The sentence 'the new outer-layer factors S_{(4)},S_{(4,1)},...,S_{(4,4)} have values 8,9,8,4,0' lists the partitions in their natural order, but the text does not say this; adding the order would avoid ambiguity.
  3. [§2.1, Eq. (2.9)] The definition of \sigma_r(g) uses wt(f)\leq r, so the term 'sparse multiple' is slightly misleading for multiples with exactly r+1 nonzero coefficients; the later text uses the same convention consistently, so this is only a terminological note.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Schur–Plücker factorization is proved in-paper; downstream GRS-count and non-GRS-density claims transparently depend on the same authors' companion preprint [12].

full rationale

The core identity (Theorem 3.3) is derived inside the paper: the remainder columns Q_i are defined by the recurrence, Lemma 3.1 gives the index-set/partition bijection, Theorem 3.2 derives the reduced-minor identity from the bialternant formula for Schur polynomials, and Proposition 2.1 re-proves the parity-check/MDS criterion. The equality Ddet = DSch is not definitional: Ddet is the determinantal product over reduced index sets and DSch is the Schur-product polynomial, and their agreement follows from the proven identity for every reduced minor. No fitted parameter is renamed as a prediction, and no input is defined in terms of the claimed output. The self-citations to [12] (GRS classification, rational-normal-curve rigidity, pure-power criterion, exact GRS count) are numerous, but the paper is explicit about its logical boundary: 'Except where we explicitly invoke [12] for the pure-power criterion and the GRS classification and counting results, the Schur–Plücker, divisor, filtration, and sparse-threshold arguments below are self-contained.' Those imported results are used only for downstream statements (Corollaries 4.10, 5.11–5.14, 6.18–6.19; Propositions 4.11, 5.13, 5.14) and do not feed back into the master Schur–Plücker theorem. Since [12] is a same-author companion preprint rather than an independently machine-checked or externally benchmarked artifact, this is a transparency and robustness caveat, not a circular reduction: no equation in the paper is equivalent by construction to its own input.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new postulated entities. It depends on standard symmetric-function facts, a standard finite-field zero bound, and several classification and counting results from the companion paper [12] by the same authors. The central Schur-Plücker derivation is self-contained.

assumptions (3)
  • standard math Standard symmetric-function facts: bialternant formula, Jacobi-Trudi identities, hook-content formula, Gaussian binomial generating functions.
    Used throughout Sections 2.5, 3, 4, and 5. These are classical results from Macdonald [16] and are not proven in the paper.
  • standard math Finite-field zero bound: a nonzero degree d polynomial over F_q has at most d q^{r-1} zeros in F_q^r.
    Used in Corollaries 4.9, 4.13, 5.3, and 5.11. This is a standard bound from Lidl and Niederreiter [13].
  • domain assumption Companion-paper inputs: rational-normal-curve rigidity, GRS classification, pure-power MDS criterion, and exact GRS counts from [12].
    These results, from Li and Yuan [12, Lemma 4.3, Theorem 5.3, Lemma 5.2, Theorem 6.14, Corollaries 5.5 and 5.6], are imported as black boxes for the GRS-type and pure-power corollaries in Sections 4.2, 4.3, 5.5, and 6.2. They are not used in the self-contained Schur-Plücker proof.

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Pith. "Pith review of Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes." pith.science (2026). https://pith.science/paper/VVJYCHT4

@misc{pith2026260801146,
  author       = {Pith},
  title        = {Pith review of: Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVJYCHT4}},
  note         = {Machine review of arXiv:2608.01146}
}
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.

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