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A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic

Xiang Fan

arXiv:2610.03740v2cs.ITmath.COmath.IT

TL;DR

The MDS conjecture remained open in general over proper extension fields, with prior general unconditional ranges on a square-root scale. This paper proves an arbitrary-field positive-characteristic rank theorem and applies it over finite fields to establish two MDS dimension ranges. For fixed odd p, those ranges cover an asymptotic proportion 1−1/(2p−3) of dimensions.

  • Problem

    The MDS conjecture was known over prime fields but remained open in general over proper extension fields.

    Source: §1 · p. 1

  • Method

    A positive-characteristic vanishing theorem gives full row rank for Chowdhury’s matrices, and specialization to F_q combines that result with the Ball–Lavrauw construction.

    Source: §1 · p. 2

  • Results

    For odd-prime-power q, the MDS conjecture holds for 2 ≤ k ≤ B(p,q) or q+2−B(p,q) ≤ k ≤ q, covering an asymptotic proportion 1−1/(2p−3) of dimensions for fixed p.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Every normal rational curve in PG(N,q) is a complete (q+1)-arc for odd prime-power q and 2 ≤ N ≤ q−2.

    Source: §1 · p. 3

    The full-row-rank guarantee is restricted to arc sizes satisfying k+n−1 ≤ m ≤ k+n−1+min{k−2,(n+1)(p−2)}.

    Source: §4.4 · p. 19

Abstract

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Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\mathbb F_q$ in every dimension $k$ satisfying \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q, \qquad B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor. \] For fixed $p$, the first interval is linear in $q$; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic $p>0$. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at $|G|=2k-3+n$ it proves Chowdhury's full-row-rank conjecture without the $q$-dependent restriction. Specialization to $\mathbb F_q$, together with the Ball--Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power $q$, every normal rational curve in $\mathrm{PG}(N,q)$ is complete for $2\leqslant N\leqslant q-2$, and every projective Reed--Solomon code of length $q+1$ and dimension $2\leqslant k\leqslant q-2$ has covering radius $q-k$.

1. Introduction

The paper proves the MDS conjecture over odd-characteristic finite fields in two dimension ranges, covering an asymptotic proportion 1−1/(2p−3) of dimensions for fixed p. Its arbitrary-field rank theorem supplies the algebraic step, while finite-field arguments yield the MDS range and a normal-rational-curve completeness result.

Introduction

  • The MDS conjecture remained open over general proper extension fields, while prior general unconditional ranges were on the square-root scale.The conjecture was known over prime fields [Bal12, Theorem 1.7].Source: §1 · p. 1

Theorem 1.1

  • For odd-prime-power q, the MDS conjecture holds when 2 ≤ k ≤ B(p,q) or q+2−B(p,q) ≤ k ≤ q.In these dimensions, the maximum size of a k-dimensional arc over F_q is q+1.Source: §1 · p. 2
  • For fixed odd p, the theorem covers an asymptotic proportion 1−1/(2p−3) of dimensions, changing the general unconditional range from square-root to linear scale.Earlier general ranges cover O_p(√q) dimensions even after adjoining their duals.Source: §1 · pp. 1–2

Proof strategy

  • A positive-characteristic vanishing theorem yields full row rank for Chowdhury’s matrices whenever k+n−1 ≤ m ≤ k+n−1+min{k−2,(n+1)(p−2)}.The rank theorem requires no finiteness assumption on the field.Source: §1 · p. 2
  • At m=2k−3+n, the theorem proves Chowdhury’s full-row-rank conjecture under k ≤ p+n(p−2), without its additional q-dependent restriction.Specializing to F_q and setting n=q−2k+5 identifies this condition with k ≤ B(p,q).Source: §1 · p. 2

Theorem 1.1

  • The one-step gain over Chowdhury’s earlier threshold can close the central pair of dimensions; for q=49 and p=7, it covers dimensions 25 and 26.Here B(7,49)=25, whereas BCh(7,49)=24.Source: §1 · p. 2

Corollaries

  • Every normal rational curve in PG(N,q) is a complete (q+1)-arc for odd prime-power q and 2 ≤ N ≤ q−2.This proves the odd-characteristic case of the classical normal-rational-curve completeness conjecture.Source: §1 · p. 3

Proof strategy

  • The algebraic sections use arbitrary positive-characteristic fields; the finite-field step uses the Ball–Lavrauw construction and code duality.Known completeness results are combined with the MDS theorem for the stated normal-rational-curve corollary.Source: §1 · p. 3

2. Inclusion operators on subsets

Section 2 establishes characteristic-p kernel identities for subset-inclusion operators and characterizes functions whose alternating sums on matchings vanish. These results provide the inclusion-map structure used in later sections.

2. Inclusion operators on subsets

  • When |X| ≥ 2a + 1, ker E_{a+1,a} = im E_{a,a+1−p}; when |X| ≥ 2a + p − 1, ker E_{a+p−1,a} = im E_{a,a−1}.These are the two kernel identities in Proposition 2.1.Source: §2 · p. 4
  • For 0 ≤ a < p − 1, d: F_a → F_{a+1} is injective whenever |X| ≥ 2a + 1.Source: §2 · p. 4
  • The subset-indicator vectors ⟨I⟩_a, for indices satisfying the stated rank condition, form a basis of the relevant free abelian group.The rank condition is i_s ≥ 2s for every s, with |I| ≤ min{a, N − a}.Source: §2 · p. 4
  • The functions whose alternating sums vanish on every a-matching are exactly the sum of the images im E_{a,b} for 0 ≤ b < a.Either choice in a matching pair cancels in the alternating sum; the reverse inclusion follows from the triangular matching functionals.Source: §2 · p. 5
  • Every function in the decomposition (2.3) is represented by inclusion images, but the sum need not be direct and its representation need not be unique.Source: §2 · p. 5

3. Matching relations on arcs

In characteristic p, this section develops a descent framework for matching relations on arcs, completing the lifting of inclusion-map relations and reducing them to contributions from smaller subsets. Under the stated hypotheses, the descent removes the boundary term and expresses G through lower-degree images.

3. Matching relations on arcs

  • Theorem 3.1 converts vector-valued matching relations into coefficient functions generated from smaller subsets.The paper identifies this as the only result from the section used in later sections.Source: §3 · p. 6

3.1. Gradings and the boundary operator.

  • The commuting boundary and inclusion maps organize the proof by tuple and subset degree, with arc independence giving exactness in positive tuple degree.For an independent set of arc vectors, the boundary complex is exact in positive tuple degree; the boundary image at degree zero is the kernel of the quotient map.Source: §3.1 · pp. 7–8

3.2. The quotient condition at tuple degree zero.

  • Kernel corrections enforce the quotient condition at tuple degree zero while preserving inclusion images and allow representations to use coefficient arrays satisfying that condition.The coefficient arrays are adjusted by elementary relations that leave the total inclusion image unchanged.Source: §3.2 · pp. 8, 10

3.3. Lifting inclusion-map relations.

  • If ∂G = dR under the lemma’s hypotheses, then G = dC + ∂Z; in particular, G = 0 when t = 1, and Z = 0 when µ = e.The proof descends in tuple degree, alternates between d and d^(p−1) on subset indices, and uses exactness to lift the relation back.Source: §3.3 · p. 11
  • Repeated applications of Lemma 3.2 lift the correction through successive subset degrees, yielding G − dC = ∂Z; the target for Z is zero when µ = e.The correction preserves d b X0 during the lifting procedure.Source: §3.3 · p. 12

3.4. Descent in subset degree.

  • Under the stated hypotheses, Lemma 3.6 produces Z and lower-degree terms for G; when µ = e, Z vanishes, and when t = 1, G = 0.In the application to Theorem 3.1, the cardinality and dimension requirements match the theorem’s assumptions, and the vanishing boundary term leaves G as a sum of images from smaller subset degrees.Source: §3.4 · pp. 12, 14
  • The descent detects the largest remaining subset degree, then lowers it until the relation has the form ∂G′ = dR′ and Lemma 3.5 applies.The detection step uses matching-supported alternating sums to annihilate contributions from smaller column sizes; terms divisible by p are retained at strictly smaller subset degree.Source: §3.4 · pp. 13–14

4. Global vanishing and determinant relations

Section 4 proves a vanishing theorem for coefficient arrays and uses projection to eliminate determinant relations, deriving full-row-rank results for Chowdhury’s matrices. It also establishes that G has full row rank and specifies when Theorem 4.4 applies to Corollary 4.5.

4.1. Vanishing of coefficient arrays.

  • The coefficient-array lemma forces the symmetric function to vanish under the stated arc size and dimension bounds and its three conditions.The proof lowers the representation degree until distinct-index coordinates vanish, then uses the first two conditions to eliminate repeated-index coordinates.Source: §4.1 · pp. 15–16

4.2. A linear system associated with an arc.

  • Under its cardinality and dimension bounds, the scalar system associated with an arc has only the zero solution.The bounds include N ≥ 2r + e and dim K* = N − r − 1 ≥ r + e − 1 + max{0, r − (e − 1)(p − 2)}.Source: §4.2 · p. 17

4.3. Projection of determinant relations.

  • Under the displayed inequality, |HD| = p + 2ρ + e − 2 satisfies Theorem 4.2’s cardinality hypothesis, hence x = 0; choosing D ⊆ B0 of size k − 1 − ρ yields a contradiction.Source: §4.3 · p. 19

4.4. Chowdhury’s matrices and full row rank.

  • For k + n −1 ≤ m ≤ k + n −1 + min{k −2, (n + 1)(p −2)}, Chowdhury’s matrix has full row rank over any field of characteristic p > 0.At m = 2k −3 + n, this proves the full-row-rank conjecture without its finite-field size restriction; for k-dimensional arcs, the corollary applies when k ≤ p + n(p−2).Source: §4.4 · pp. 19–20
  • At m = 2k-3+n, G has full row rank when k ≤ p+n(p-2).This condition is equivalent to k-2 ≤ (n+1)(p-2), so Theorem 4.4 applies.Source: §4.4 · p. 20
  • For Corollary 4.5, the condition at m = 2k-3+n reduces to k ≤ p+n(p-2), allowing Theorem 4.4 to apply.Equivalently, k-2 ≤ (n+1)(p-2).Source: §4.4 · p. 20

5. Finite-field deduction of the MDS bound

The finite-field argument turns a hypothetical (q + 2)-point arc into a nonzero left-kernel vector, contradicting the full-row-rank theorem; duality then gives the second dimension interval.

5. Finite-field deduction of the MDS bound

  • The finite-field deduction specializes the arbitrary-positive-characteristic rank theorem to F_q and uses the contradiction between the constructed kernel vector and full row rank.For k ≥ 3, the interval condition ensures e ≥ 0, matching the parameter range used for the matrix.Source: §5 · pp. 20–21
  • The Ball–Lavrauw construction gives M↑e a nonzero left-kernel vector from a hypothetical (q + 2)-point arc, with every coordinate nonzero.The construction uses t = k − 3 and n = e; its all-nonzero vector remains a left-kernel vector after rescaling representatives.Source: §5 · pp. 20–21
  • The standard q + 1 vectors form a k-dimensional arc, so the proof establishes the upper bound by ruling out a (q + 2)-point arc.The arc uses (1, t, t^2, …, t^(k−1)) for t ∈ F_q together with (0, …, 0, 1).Source: §5 · p. 21
  • A hypothetical (q + 2)-point arc contradicts full row rank when 2 ≤ k ≤ B(p, q), and duality excludes such arcs for q + 2 − B(p, q) ≤ k ≤ q.For k ≥ 3, set e = q − 2k + 5; then q + 2 = 2k − 3 + e and k ≤ B(p, q) is equivalent to k ≤ p + e(p − 2).Source: §5 · p. 21

6. Applications to normal rational curves and projective Reed–Solomon codes

The paper proves completeness of normal rational curves throughout the stated odd-field range by joining low- and high-dimensional bounds. This also gives the covering-radius result for projective Reed–Solomon codes.

6. Applications to normal rational curves and projective Reed–Solomon codes

  • The completeness proof joins Storme’s lower-dimensional range to the MDS-theorem upper-dimensional range, with Lemma 6.1 ensuring the ranges meet.For odd nonprime q ≠ 27, t(q) ≤ B(p, q) + 1; prime fields and q = 27 are handled separately by cited results.Source: §6 · p. 22
  • For odd nonprime q ≠ 27, the proof combines Storme’s completeness criterion with the bound and computer-search data for t(q).The low-dimensional range is controlled by t(q), while Theorem 1.1 supplies the upper-dimensional range.Source: §6 · p. 22
  • Every normal rational curve in PG(N, q) is complete for odd prime powers q and 2 ≤ N ≤ q − 2.A normal rational curve is a (q + 1)-arc, and completeness means no further point can be adjoined while preserving the arc property.Source: §6 · pp. 21–22
  • Every projective Reed–Solomon code of length q + 1 and dimension 2 ≤ k ≤ q − 2 has covering radius q − k.The proof applies Dür’s equivalence between this covering radius and completeness of the degree-(q − k) normal rational curve.Source: §6 · pp. 21–22

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