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On the Length of Maximum Sum-Rank Distance Codes

Valentina Astore, Ferdinando Zullo

arXiv:2610.23633v1cs.ITmath.COmath.IT

TL;DR

The paper asks how the arc-based geometric approach to MDS-code length extends to MSRD codes, whose block profiles add constraints beyond total length. It studies MSRD systems through their linear-set geometry and derives length bounds in dimension three and arbitrary dimension. The results include a full-length-block bound and structural restrictions on MSRD codes.

  • Problem

    The paper asks how the geometric arc and MDS length theory extends to MSRD codes, where block distribution matters alongside total length.

    Source: §1 · p. 3

  • Method

    The paper studies MSRD systems through their linear-set geometry, interpreting shortening as projection to derive length bounds.

    Source: §1 · p. 4

  • Results

    For full-length blocks, the number of blocks is at most q − 1 whenever κ ≤ m, and exceptional configurations can occur.

    Source: §7 · p. 26

  • Takeaways & Limitations

    Every MSRD code is MDS over F_Q in the Hamming metric, but its block profile imposes additional geometric restrictions.

    Source: §7 · p. 26

    Scattered, hyperplane-scattered, pairwise-disjoint component linear sets need not make a system MSRD.

    Source: §3 · p. 10

Abstract

from arXiv · show

The classical theory of arcs in finite projective spaces provides a powerful geometric framework for studying maximum distance separable (MDS) codes and their possible lengths. In this paper, we investigate how this perspective extends to maximum sum-rank distance (MSRD) codes, whose geometric description involves systems of linear sets rather than individual projective points. Our main goal is to understand how the block structure of an MSRD code constrains its length. In dimension three, we determine the possible intersection patterns of lines with the associated linear sets and obtain explicit formulas for their characters and the weight distribution of the corresponding codes. These results lead to new length bounds and show that the block structure comes at a cost: the presence of a block of rank at least two reduces the maximum possible total length by at least $q-1$, and sufficiently long MSRD codes must be classical, that is, MDS codes over $\mathbb F_{q^m}$. We also prove that three-dimensional one-weight MSRD codes do not exist. In arbitrary dimension, we interpret shortening geometrically as projection and establish a general projection bound that extends classical counting arguments for arcs. This yields restrictions depending explicitly on the block profile and, for full-length blocks, identifies parameter ranges in which linearized Reed--Solomon codes attain the maximum possible number of blocks. Finally, we construct an exceptional three-dimensional MSRD code over $\mathbb F_9$ with three blocks of length two and propose a sum-rank analogue of the MDS conjecture.

1 Introduction

The paper asks how the geometric theory of arcs and MDS-code lengths extends to MSRD codes, where block profiles matter in addition to total length. It develops geometric bounds and structural results in dimension three and arbitrary dimension.

1 Introduction

  • Arc geometry connects MDS codes to finite geometry: a code’s generator-matrix columns form an arc exactly when the code is MDS.The paper frames MSRD codes as a setting in which to extend this correspondence.Source: §1 · p. 2
  • The sum-rank metric partitions coordinates into blocks and sums their F_q-ranks, extending both Hamming and rank metrics.The block distribution affects MSRD length constraints, not just total length.Source: §1 · p. 3
  • In dimension three, the paper determines line-intersection sizes and character and weight distributions, deriving length restrictions from lines through a point.A block of rank at least two reduces the maximum total length by at least q − 1; sufficiently long codes, or codes with sufficiently many blocks, must be classical MDS codes over F_Q.Source: §1 · pp. 3–4
  • The paper proves that three-dimensional one-weight MSRD codes do not exist.Source: §1 · p. 4
  • In arbitrary dimension, shortening becomes projection geometrically, yielding a bound that extends arc-counting arguments and constrains block profiles.For full-length blocks, the paper identifies parameter ranges where the bound t ≤ q − 1 holds and is attained by linearized Reed–Solomon codes; it also gives an exceptional F_9 construction and proposes an MSRD analogue of the MDS conjecture.Source: §1 · p. 4

2 Preliminaries

The preliminaries establish the finite-projective-space and coding-theory language used to describe MSRD systems. They relate arc bounds to MDS lengths and introduce the block-profile geometry underlying the later results.

2 Preliminaries

  • With q a prime power and Q := q^m, PG(k−1,q) consists of one-dimensional subspaces of F_q^k; an N-arc has no k points on a hyperplane, and N ≤ q+k−1.Source: §2 · p. 4, §2.1 · p. 5

2.2 Linear codes and MDS codes

  • A linear code is MDS exactly when its projective system forms an arc, linking maximum code length to the maximum size of an arc.For odd q, a [q + 1, 3, q − 1]_q MDS code has a conic projective system and is monomially equivalent to an extended Reed–Solomon code.Source: §2.2 · pp. 6–7

2.3 Sum-rank metric codes and their geometry

  • The sum-rank weight sums the F_q-ranks of coordinate blocks; one-coordinate blocks recover Hamming weight, and a single block recovers rank weight.MSRD codes attain the Singleton-like bound d_SR ≤ N − k + 1; the profile records block lengths, and the associated system is a tuple of F_q-subspaces.Source: §2.3 · pp. 7–8
  • A code’s block subspaces define a projective MSRD system of linear sets, with the MSRD condition characterized by their hyperplane weights.Linearized Reed–Solomon codes are MSRD, and their associated linear sets lie on a q-analogue of the normal rational curve.Source: §2.3 · pp. 8–9

3 First properties of the MSRD condition

The first properties show that MSRD systems impose strong intersection restrictions, making their block vectors an arc while retaining block-specific constraints. Classical arc bounds control total length, but the profile yields additional structure.

3 First properties of the MSRD condition

  • Every component linear set is h-scattered for h < k, and the component linear sets are pairwise disjoint.These properties are necessary but not sufficient for the system to be MSRD: four disjoint singleton linear sets can still have three points on one line, giving that line weight 3.Source: §3 · p. 10
  • Choosing an F_q-basis from each block of an MSRD system gives N distinct projective points forming an N-arc of PG(k − 1,Q).Consequently, arc-size bounds apply directly to total MSRD length.Source: §3 · p. 10
  • For k = 3, the total length satisfies N ≤ Q + 2, or N ≤ Q + 1 when Q is odd.When the MDS conjecture holds over F_Q, a non-trivial system has N ≤ Q + 1 except for its stated even-field exceptions.Source: §3 · p. 10
  • For dimension two, the MSRD condition is equivalent to scattered, pairwise disjoint linear sets in PG(1,Q), so their total number of points is at most Q + 1.Equality occurs for extended Reed–Solomon codes with singleton blocks and for profile (m, …, m, 1, 1) with q − 1 blocks of rank m.Source: §3 · p. 11

4 Three-dimensional MSRD codes

In dimension three, the MSRD condition restricts line intersections with the union of associated linear sets to four sizes, enabling explicit character and weight-distribution formulas. These geometric constraints yield bounds on total length and block count, and force sufficiently long systems to be classical.

4 Three-dimensional MSRD codes

  • Only 0, 1, 2, or q + 1 points can lie where a line meets the union of the linear sets.These intersection sizes correspond respectively to external lines, tangents, 2-secants, and (q + 1)-secants.Source: §4 · p. 11
  • The character formulas determine a1 and a0 from the total point count M and the numbers of 2- and (q + 1)-secants.The results also give a2 as the sum of products of point counts from distinct linear sets, and aq+1 from the blocks.Source: §4 · p. 12
  • The three-dimensional code’s weight distribution depends only on q, m, and its block profile.The nonzero weights are N − 2, N − 1, and N, with multiplicities determined by the corresponding line characters.Source: §4 · p. 13
  • N ≤ Q + 2 for every three-dimensional MSRD system; if it is nonclassical with t ≥ 2, then N ≤ Q + 3 − q.A rank-one block gives the sharper bound N ≤ Q + 2 − (q − 1)e, where e = N − t.Source: §4 · p. 14
  • A block of rank at least two reduces the maximum possible total length by at least q − 1.For a rank-one block, the bound can be written t + qe ≤ Q + 2.Source: §4 · pp. 14–15
  • At total length Q + 2, the system is a classical hyperoval and Q is even.When q is odd and N = Q + 1, the system is classical and its point set is a conic.Source: §4 · p. 15
  • If t ≥ Q + 2 − q, the system is classical and its point set is a t-arc in PG(2, Q).The boundary case is excluded by a parity argument.Source: §4 · pp. 15–16

5 One-weight MSRD codes in dimension three

A three-dimensional one-weight MSRD code would require every projective line to have total weight two, leaving no external or tangent lines. The paper proves that this configuration cannot occur for any field or block profile.

5 One-weight MSRD codes in dimension three

  • Such a code would require every line of PG(2, Q) to have weight exactly two, so there could be no external or tangent lines.Each line would be either a (q + 1)-secant to one linear set or a 2-secant meeting two distinct linear sets.Source: §5 · p. 16
  • A hypothetical three-dimensional one-weight MSRD system must have equal block lengths.The common block length follows because the total point count is independent of which block contains a chosen point.Source: §5 · p. 16
  • No non-degenerate three-dimensional one-weight MSRD code exists over F_Q.The nonexistence result applies regardless of field or block profile.Source: §5 · pp. 16–17

6 MSRD codes of arbitrary dimension

This section studies length and block-count bounds for arbitrary-dimensional MSRD systems using geometric shortening as projection and constraints from the block profile and dimension. It also presents exceptional full-block systems and proposes a general upper-bound conjecture for realizable profiles.

6.1 Shortening, puncturing and duality

  • Shortening on j ≤ k−2 coordinates projects the system and yields a non-degenerate MSRD code of dimension k−j with unchanged minimum sum-rank distance.Geometrically, the projection is from the subspace generated by the shortened coordinates.Source: §6.1 · pp. 18–19
  • Puncturing one coordinate preserves dimension and, when N ≥ k+1, the MSRD property while decreasing minimum sum-rank distance by one; duality preserves the profile.The dual is MSRD when k ≤ N−1.Source: §6.1 · pp. 18–19

6.2 General bounds

  • Projecting an MSRD system to dimension two shows that every hyperplane through the shortening subspace has weight k−2 or k−1, and the theorem counts those attaining each weight.The projection is obtained by shortening k−2 suitably chosen dimensions.Source: §6.2 · p. 19
  • A minimum-loss shortening distribution shortens whole blocks except possibly one partially, and λ_j(n) ≥ j, with equality exactly when at least j blocks have rank one.The loss minimization identifies the strongest bound obtainable by distributing a fixed number of shortened dimensions among blocks.Source: §6.2 · p. 20
  • For κ ≥ 3, if some block i has n_i ≥ κ − 2, then M(n) ≤ Q + 1 + q^(n_i−κ+2)θ_(κ−3).Source: §6.2 · p. 22
  • If a block has rank at least two and the remaining blocks total at least κ−2, then N ≤ Q + κ−q; if at least κ−2 blocks have rank one, then t + q(N−t) ≤ Q + κ−1.These bounds explicitly constrain length and block profile.Source: §6.2 · p. 22
  • For non-classical systems, t ≤ Q + κ − q − 1.Source: §6.2 · p. 23

6.3 Equal and full blocks

  • Full-block systems satisfy t ≤ q − 1 when κ ≤ m and t ≤ q when κ = m + 1; linearized Reed–Solomon codes attain t = q − 1 in every dimension.Source: §6.3 · pp. 23–24

6.4 The exceptional full-block case

  • The q-block full-length case is impossible for q = 2, but for (q, m) = (3, 2) a three-dimensional MSRD code exists with three blocks of length two.Source: §6.4 · pp. 24–25
  • If a q-block system as posed exists, every (m − 1)-dimensional subspace of a block lies on exactly one hyperplane of weight m − 1; for m = 2, each point lies on exactly one tangent line.Source: §6.4 · p. 24

6.5 An MSRD conjecture

  • For κ ≥ 4, the MSRD conjecture predicts M(n) ≤ Q + 1, implying t ≤ q − 1 for full blocks; the κ = 3 hypothesis is necessary.For κ = 3, hyperovals can have M(n) = Q + 2, and the constructed system has M(n) = 12 = Q + 3.Source: §6.5 · p. 26

7 Conclusions

The paper shows that MSRD length is constrained by geometric restrictions that depend on the block profile. It derives dimension-three results and arbitrary-dimensional projection bounds, including constraints for full-length blocks and a conjectured sum-rank analogue of the MDS conjecture.

7 Conclusions

  • In dimension three, line-intersection sizes and character distributions yield total-length bounds, imply sufficiently long MSRD codes are classical, and rule out one-weight MSRD codes.The paper also states that sufficiently many blocks force codes to be classical.Source: §7 · p. 26
  • For full-length blocks, their number is at most q −1 when κ ≤m, matching linearized Reed–Solomon parameters; an explicit construction gives an exceptional configuration, and the paper conjectures M(n) ≤Q + 1 when κ ≥4.Whether further exceptions exist, particularly in the range of Problem 6.16, remains open.Source: §7 · p. 26

Statements and Declarations

The authors report funding support for both authors, no relevant competing interests, and no data sharing because no datasets were generated or analyzed.

Statements and Declarations

Declaration of Generative AI and AI-assisted Technologies in the Writing Process

AI-assisted tools were used only for language editing and improving manuscript clarity. The authors say the mathematical work is theirs and that they reviewed and take responsibility for the manuscript.

Declaration of Generative AI and AI-assisted Technologies in the Writing Process

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