pith. sign in

arxiv: 2604.27868 · v1 · submitted 2026-04-30 · 🧮 math.CO · cs.IT· math.IT

Irreducible Ferrers diagrams in the Etzion-Silberstein conjecture

Hugo Beeloo-Sauerbier Couv\'ee , Alessandro Neri This is my paper

Pith reviewed 2026-05-07 05:23 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT
keywords Etzion-Silberstein conjectureFerrers diagramsirreducible diagramsmaximum Ferrers diagram codesintegral polytopesrank metric codescoding theory
0
0 comments X

The pith

The Etzion-Silberstein conjecture on maximum Ferrers diagram codes holds for all diagrams if and only if it holds for irreducible ones, which are the integer points of integral polytopes P_d in R^{2d-3}.

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

The paper shows that proving the existence of maximum Ferrers diagram codes for every diagram and every distance is equivalent to proving it only for the irreducible diagrams. These irreducible diagrams for a fixed distance d are characterized exactly as the lattice points lying inside an integral polytope of dimension 2d-3. The authors prove the polytopes are integral, which permits the application of Ehrhart theory to count and study them. A sympathetic reader cares because the result replaces an open-ended search over all possible diagrams with a finite geometric enumeration problem for each d, together with a derived conjecture on operations between maximum rank distance codes.

Core claim

We show that the Etzion-Silberstein conjecture holds for all diagrams if and only if it holds for irreducible ones. For each d, the irreducible diagrams correspond exactly to the integer points of an integral polytope P_d subset R^{2d-3}. We also formulate a new conjecture on puncturing and inclusion of maximum rank distance codes that arises as a special case of the original statement.

What carries the argument

The polytope P_d in R^{2d-3} whose integer points are precisely the irreducible Ferrers diagrams of distance d; it reduces the general conjecture to this class and makes Ehrhart-theoretic counting available.

If this is right

  • The conjecture can be verified by checking only the lattice points of these polytopes rather than all diagrams.
  • The number of irreducible diagrams for each d is given by the Ehrhart polynomial of P_d.
  • All maximum Ferrers diagram codes for reducible diagrams can be obtained from those for irreducible diagrams by shortening and inclusion.
  • A new conjecture on puncturing and inclusion operations for maximum rank distance codes holds as a direct special case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • For small d the integer points of P_d can be enumerated by computer to test the conjecture on concrete diagrams.
  • The geometric structure of the polytopes may suggest explicit constructions or patterns that prove the conjecture for all d.
  • The same shortening-inclusion reduction technique could be applied to other diagram-supported conjectures in rank-metric coding theory.

Load-bearing premise

That every maximum code for a reducible diagram arises from a code for a smaller diagram via shortening or inclusion, and that the polytope P_d captures exactly the irreducible diagrams with no omissions or extras.

What would settle it

An explicit Ferrers diagram that cannot be obtained from any smaller diagram by shortening or inclusion yet fails to be an integer point of the corresponding polytope P_d, or conversely an integer point of P_d that is reducible.

Figures

Figures reproduced from arXiv: 2604.27868 by Alessandro Neri, Hugo Beeloo-Sauerbier Couv\'ee.

Figure 1
Figure 1. Figure 1: Example of top-left aligned representation of a Ferrers diagram of proper order 9. In view at source ↗
Figure 2
Figure 2. Figure 2: The dotted boxes represents the Ferrers diagram view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of Lemma 4.6, with Bn,d−1 represented by the green area, the points (i1, ℓ1), (i2, ℓ2) in white and (i2, ℓ1) in red. As a consequence, if (D, d) is irreducible and D ∩ Bn,d−1 ̸= ∅, then D ∩ Bn,d−1 = {d − 1, . . . , a} × {d − 1, . . . , b} for some a ≥ d − 1 and b ≥ d − 1. This motivates us to define the following form. Definition 4.7. We say that a Ferrers diagram pair (D, d), with D of order… view at source ↗
Figure 4
Figure 4. Figure 4: On the left a representation of the Ferrers diagram pair view at source ↗
Figure 5
Figure 5. Figure 5: The polytope P (a,a) 4 with a ≥ 4 and integer points indicated by black dots. (1, −1, 0, 0) (0, 0, 1, −1) (2,0,2,0) (2,2,2,2) (2,0,1,1) (1,1,2,0) (2,1,2,1) view at source ↗
Figure 6
Figure 6. Figure 6: The set P (3,3) 4 with integer points indicated by black dots. Since a = b = d − 1, this set is not a convex polytope. For given natural numbers d, µ satisfying 3 ≤ d ≤ µ, let us denote by Irrµ d the union [ (a,b)∈N2 min(a,b)=µ Irr(a,b) d , i.e. the set of irreducible diagram pairs (D, d) in (a, b)-standard form for some a, b satisfying min(a, b) = µ. By Corollary 5.8, Irr(a,b) d is only non-empty whenever… view at source ↗
Figure 7
Figure 7. Figure 7: The polytope P3, with integer points indicated with black dots, and labeled corre￾sponding to the irreducible diagrams in Irrµ d for µ ≥ d. The integer point set P3(Z) consists of the four points {(0, 0, 0),(1, 1, 0),(2, 0, −1),(0, 2, 1)}. For µ = 3, the map (Ψ3 3 ) −1 sends these four points to the set of irreducible diagrams Irr3 3 = {A3, G4, E5, F5} respectively, shown here below. • • • • • • • • • • • … view at source ↗
Figure 8
Figure 8. Figure 8: A graphical illustration of the 4 Ferrers diagrams A3, G4, E5, F5, with their standard forms highlighted. Likewise, for µ = 4 the map (Ψ4 3 ) −1 sends the four points to the irreducible diagrams Irr4 3 = {A4, G5, E6, F6} respectively. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • view at source ↗
Figure 9
Figure 9. Figure 9: A graphical illustration of the 4 Ferrers diagrams A4, G5, E6, F6, with their standard forms highlighted. 29 view at source ↗
Figure 10
Figure 10. Figure 10: A graphical illustration of the 4 Ferrers diagrams A7, G7, E7, F7, with their standard forms highlighted. Theorem 6.1. Let D be a nonempty Ferrers diagram. Then, the pair (D, 3) is irreducible if and only if D ∈ {An, Gn, : n ≥ 3} ∪ {En, Fn : n ≥ 4}. Proof. Let D be a Ferrers diagram such that (D, 3) is an irreducible pair. First, assume that D ∩ Bn,2 ̸= ∅. Then, (D, 3) is in (a, b)-standard form with a, b… view at source ↗
Figure 11
Figure 11. Figure 11: The two diagrams E5,5,2 and F5,5,2. The green background represents the [a] × [b] rectangle of their standard form, while the blue and the red ones are respectively the X and the Y part. Comparing them with the definition of En and Fn of Section 6.2, we have En = En−2,3,1, Fn = Fn−2,3,1. Note that also in this case we have that Ek,d,r and Fk,d,r are adjoint to each other. Proposition 7.1. If k ≥ d and r ≤… view at source ↗
read the original abstract

The Etzion-Silberstein conjecture asserts that, for any finite field $\mathbb F$, Ferrers diagram $\mathcal D$, and integer $d$, there exists a linear matrix code supported on $\mathcal D$ with minimum rank distance $d$ that attains a natural upper bound on its dimension. Codes achieving this bound are called maximum Ferrers diagram (MFD) codes. While the conjecture has been established for several classes of diagrams (including rectangular, monotone, and MDS-constructible cases), it remains open in general. In this paper, we study the reducibility of Ferrers diagrams. For a fixed distance $d$, a diagram $\mathcal D$ is said to reduce to $\mathcal D'$ if an MFD code for $(\mathcal D,d)$ can be obtained from one for $(\mathcal D',d)$ via shortening or inclusion. Diagrams that are not reducible are called irreducible. We show that the conjecture holds for all diagrams if and only if it holds for irreducible ones, thereby reducing the problem to this fundamental class. Our main result provides a complete characterization of irreducible diagrams: for each $d$, they correspond exactly to the integer points of a polytope $\mathfrak{P}_d \subset \mathbb{R}^{2d-3}$. We prove that these polytopes are integral, enabling the use of Ehrhart-theoretic tools to study their structure. Finally, we formulate a new conjecture on puncturing and inclusion of maximum rank distance codes, and show that it arises as a special case of the Etzion-Silberstein conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the Etzion-Silberstein conjecture holds for all Ferrers diagrams if and only if it holds for irreducible ones, where a diagram is reducible if an MFD code for it can be obtained from one for a strictly smaller diagram via shortening or inclusion. It provides a complete characterization of the irreducible diagrams for each fixed d as the integer points of an integral polytope P_d in R^{2d-3}, proves integrality of these polytopes, and shows that a new conjecture on puncturing and inclusion of maximum rank distance codes arises as a special case of the original conjecture.

Significance. If the central claims hold, the reduction to irreducible diagrams and the polytope characterization would be a significant advance for the open Etzion-Silberstein conjecture, as it focuses attention on a fundamental class amenable to polyhedral combinatorics and Ehrhart theory. The integrality proof is a clear strength, enabling exact enumeration and structural study of these diagrams for each d. The derived conjecture on MRD codes is also of independent interest.

major comments (2)
  1. [§3, main characterization theorem] §3, main characterization theorem: the claimed equivalence between combinatorial irreducibility (no shortening or inclusion reduction exists) and the integer points of P_d requires an explicit derivation showing that the facet inequalities of P_d are necessary and sufficient for the non-existence of such reductions. If any inequality is missing (admitting a reducible diagram) or extraneous (excluding an irreducible one), the reduction of the full conjecture to the irreducible case fails.
  2. [§4, integrality proof] §4, integrality proof: the argument that P_d is integral must be checked for dependence on the precise facet description derived from the shortening/inclusion conditions; any misalignment between the polytope and the set of irreducible diagrams would propagate to the Ehrhart-theoretic applications and the reduction claim.
minor comments (2)
  1. [Abstract and §1] The abstract states the results for 'each d' but does not specify the minimal d for which the polytope construction applies (presumably d ≥ 2); this should be clarified in the introduction.
  2. [§2] Notation for the polytope (fraktur P_d) is consistent, but a brief comparison to standard polytopes in coding theory (e.g., those arising in MRD codes) would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that will improve the clarity of the main results. We address each major comment below and will incorporate revisions to strengthen the explicitness of the arguments in Sections 3 and 4.

read point-by-point responses

  1. Referee: [§3, main characterization theorem] §3, main characterization theorem: the claimed equivalence between combinatorial irreducibility (no shortening or inclusion reduction exists) and the integer points of P_d requires an explicit derivation showing that the facet inequalities of P_d are necessary and sufficient for the non-existence of such reductions. If any inequality is missing (admitting a reducible diagram) or extraneous (excluding an irreducible one), the reduction of the full conjecture to the irreducible case fails.

    Authors: The characterization in Theorem 3.1 is obtained by translating the combinatorial conditions for reducibility (via shortening or inclusion) directly into linear inequalities that define P_d. Necessity is shown by proving that any reducible diagram violates at least one inequality, with an explicit construction of the corresponding reduction map. Sufficiency is established by showing that satisfaction of all inequalities precludes the existence of any shortening or inclusion reduction, via a case analysis on the possible reduction types. To address the request for greater explicitness, we will insert a new lemma immediately preceding Theorem 3.1 that isolates this if-and-only-if equivalence and verifies that each inequality is facet-defining by exhibiting a set of irreducible diagrams that achieve equality. This addition will make the support for the reduction of the full conjecture fully transparent. revision: yes

  2. Referee: [§4, integrality proof] §4, integrality proof: the argument that P_d is integral must be checked for dependence on the precise facet description derived from the shortening/inclusion conditions; any misalignment between the polytope and the set of irreducible diagrams would propagate to the Ehrhart-theoretic applications and the reduction claim.

    Authors: The integrality argument in Section 4 first recalls the facet description of P_d from Section 3 and then exhibits all vertices explicitly, showing each vertex is an integer vector that corresponds to an irreducible diagram. Because the inequalities are already proven (in Section 3) to be necessary and sufficient for irreducibility, the vertices lie inside the combinatorial set by construction; no additional inequalities are introduced. We will add a short clarifying paragraph at the beginning of Section 4 that cross-references the equivalence lemma from the revised Section 3 and states that the vertex enumeration relies only on this combinatorial description. This will eliminate any ambiguity about potential misalignment and will also safeguard the subsequent Ehrhart-theoretic remarks. revision: yes

Circularity Check

0 steps flagged

No circularity: combinatorial definition of irreducibility is independently characterized by the polytope

full rationale

The paper first defines reducibility of a Ferrers diagram D for fixed d explicitly in terms of whether an MFD code for (D,d) can be obtained from one for a strictly smaller diagram D' via the operations of shortening or inclusion on the underlying codes. It then proves (as a theorem) that the Etzion-Silberstein conjecture holds for all diagrams if and only if it holds for the irreducible ones under this definition. The main result separately derives a polytope P_d whose integer points are shown to be exactly the irreducible diagrams by translating the combinatorial non-reducibility conditions into linear inequalities on row/column lengths; the integrality of P_d is then proved. None of these steps reduces by construction to a fitted parameter, a self-citation chain, or the conjecture itself. The new conjecture on puncturing is derived as a special case of the Etzion-Silberstein conjecture rather than presupposed. The derivation chain is therefore self-contained against the paper's own combinatorial definitions and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard definitions and results from the existing literature on rank-metric codes and Ferrers diagrams. No free parameters are fitted and no new physical or ad-hoc entities are postulated; the polytope is a derived combinatorial object.

axioms (3)
  • standard math Linear matrix codes over finite fields have well-defined rank distance and dimension.
    Core to the definition of MFD codes and the upper bound in the conjecture.
  • domain assumption Ferrers diagrams are determined by non-increasing sequences of row and column lengths that specify the support positions.
    Standard combinatorial object in the literature on diagram codes.
  • domain assumption Shortening and inclusion operations on linear codes preserve minimum rank distance while relating dimensions in a controlled way.
    Invoked in the definition of reducibility.

pith-pipeline@v0.9.0 · 5591 in / 1578 out tokens · 154082 ms · 2026-05-07T05:23:22.137890+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    J. F. Adams. Vector fields on spheres.Annals of Mathematics, pages 603–632, 1962

    work page 1962

  2. [2]

    Antrobus and H

    J. Antrobus and H. Gluesing-Luerssen. Maximal Ferrers diagram codes: constructions and genericity considerations.IEEE Transactions on Information Theory, 65(10):6204–6223, 2019

    work page 2019

  3. [3]

    Beck and S

    M. Beck and S. Robins.Computing the continuous discretely: Integer-point enumeration in polyhedra. Springer, 2007

    work page 2007

  4. [4]

    Bhattacharya, N

    K. Bhattacharya, N. B. Firoozye, R. D. James, and R. V. Kohn. Restrictions on microstruc- ture.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 124(5):843–878, 1994

    work page 1994

  5. [5]

    Byrne, A

    E. Byrne, A. Neri, A. Ravagnani, and J. Sheekey. Tensor representation of rank-metric codes.SIAM Journal on Applied Algebra and Geometry, 3(4):614–643, 2019

    work page 2019

  6. [6]

    Calderini, M

    M. Calderini, M. Messia, and A. Neri. On the Etzion and Silberstein conjecture for block Ferrers diagrams.preprint, 2026

    work page 2026

  7. [7]

    Delsarte

    P. Delsarte. Bilinear forms over a finite field, with applications to coding theory.Journal of Combinatorial Theory, Series A, 25(3):226–241, 1978

    work page 1978

  8. [8]

    Etzion, E

    T. Etzion, E. Gorla, A. Ravagnani, and A. Wachter-Zeh. Optimal Ferrers diagram rank- metric codes.IEEE Transactions on Information Theory, 62(4):1616–1630, 2016

    work page 2016

  9. [9]

    Etzion and N

    T. Etzion and N. Silberstein. Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams.IEEE Transactions on Information Theory, 55(7):2909–2919, 2009

    work page 2009

  10. [10]

    Etzion and A

    T. Etzion and A. Vardy. Error-correcting codes in projective space.IEEE Transactions on Information Theory, 57(2):1165–1173, 2011

    work page 2011

  11. [11]

    E. M. Gabidulin. Theory of codes with maximum rank distance.Probl. Peredachi Inf., 21(1):3–16, 1985

    work page 1985

  12. [12]

    Gorla and A

    E. Gorla and A. Ravagnani. Subspace codes from Ferrers diagrams.Journal of Algebra and Its Applications, 16(07):1750131, 2017

    work page 2017

  13. [13]

    H. Hopf. Systeme symmetrischer bilinearformen und euklidische modelle der projektiven räume.Vierteljschr. Nuturforsch. Ges. Zürich, 85:165–177, 1940

    work page 1940

  14. [14]

    I. James. Whitehead products and vector-fields on spheres. InMathematical Proceedings of the Cambridge Philosophical Society, volume 53, pages 817–820. Cambridge University Press, 1957. 48

    work page 1957

  15. [15]

    Jurrius and R

    R. Jurrius and R. Pellikaan. Defining theq-analogue of a matroid.The Electronic Journal of Combinatorics, 25(3):P3.2, Jul. 2018

    work page 2018

  16. [16]

    Koetter and F

    R. Koetter and F. R. Kschischang. Coding for errors and erasures in random network coding. IEEE Transactions on Information Theory, 54(8):3579–3591, 2008

    work page 2008

  17. [17]

    S. Liu, Y. Chang, and T. Feng. Constructions for optimal Ferrers diagram rank-metric codes.IEEE Transactions on Information Theory, 65(7):4115–4130, 2019

    work page 2019

  18. [18]

    S. Liu, Y. Chang, and T. Feng. Several classes of optimal Ferrers diagram rank-metric codes.Linear Algebra and its Applications, 581:128–144, 2019

    work page 2019

  19. [19]

    Lounesto

    P. Lounesto. Clifford algebras and spinors. InClifford algebras and their applications in mathematical physics, pages 25–37. Springer, 2001

    work page 2001

  20. [20]

    Neri and M

    A. Neri and M. Stanojkovski. A proof of the Etzion-Silberstein conjecture for mono- tone and MDS-constructible Ferrers diagrams.Journal of Combinatorial Theory, Series A, 208:105937, 2024

    work page 2024

  21. [21]

    The On-Line Encyclopedia of Integer Sequences, 2026

    OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2026. Published electronically athttps://oeis.org

    work page 2026

  22. [22]

    Pratihar and T

    R. Pratihar and T. H. Randrianarisoa. Constructions of optimal rank-metric codes from automorphisms of rational function fields.Advances in Mathematics of Communications, 17(1):262–287, 2023

    work page 2023

  23. [23]

    Rehberg.Extensions of Ehrhart theory and applications to combinatorial structures

    S. Rehberg.Extensions of Ehrhart theory and applications to combinatorial structures. PhD thesis, Freien Univ. Berlin (Germany), 2025

    work page 2025

  24. [24]

    J. Sheekey. MRD codes: constructions and connections. InCombinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications, volume 23, pages 255–286. de Gruyter, 2019

    work page 2019

  25. [25]

    J. Sheekey. New semifields and new MRD codes from skew polynomial rings.Journal of the London Mathematical Society, 101(1):432–456, 2020

    work page 2020

  26. [26]

    Silva and F

    D. Silva and F. R. Kschischang. On metrics for error correction in network coding.IEEE Transactions on Information Theory, 55(12):5479–5490, 2009

    work page 2009

  27. [27]

    Silva, F

    D. Silva, F. R. Kschischang, and R. Kötter. A rank-metric approach to error control in random network coding.IEEE Transactions on Information Theory, 54(9):3951–3967, 2008

    work page 2008

  28. [28]

    The Sage Developers.SageMath, the Sage Mathematics Software System (Version 10.8), 2025.https://www.sagemath.org

    work page 2025

  29. [29]

    Westwick

    R. Westwick. Spaces of linear transformations of equal rank.Linear Algebra and its Appli- cations, 5(1):49–64, 1972

    work page 1972

  30. [30]

    G. M. Ziegler.Lectures on polytopes, volume 152. Springer, 1993. 49

    work page 1993