Semidefinite and linear programming bounds for sum-rank-metric codes and non-existence results

Aida Abiad; Antonina P. Khramova; Ferdinando Zullo; Sven C. Polak

arxiv: 2604.27909 · v1 · submitted 2026-04-30 · 💻 cs.IT · math.CO· math.IT

Semidefinite and linear programming bounds for sum-rank-metric codes and non-existence results

Aida Abiad , Antonina P. Khramova , Sven C. Polak , Ferdinando Zullo This is my paper

Pith reviewed 2026-05-07 06:33 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT

keywords sum-rank metricsemidefinite programminglinear programming boundscoding theorynon-existence resultsassociation schemes

0 comments

The pith

A semidefinite programming bound can give tighter limits on the largest sum-rank-metric codes than previous methods.

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

This paper develops upper bounds on the maximum size of a code with prescribed minimum distance in the sum-rank metric, a distance that unifies the Hamming metric on vectors and the rank metric on matrices. The central contribution is a semidefinite programming relaxation whose value, in computational tests on concrete parameters, exceeds the strength of earlier linear programming and combinatorial bounds. The authors also prove that the classical Delsarte linear programming bound coincides with a recent eigenvalue bound in extremal regimes of the metric, then apply the collection of bounds to rule out the existence of certain optimal and perfect codes. A reader cares because the sum-rank metric appears in network coding and distributed storage, where knowing the exact maximum code size directly limits achievable reliability and efficiency.

Core claim

We obtain a semidefinite programming bound on the size of sum-rank-metric codes that computational experiments indicate can be strictly stronger than prior bounds. We establish the equivalence of the Delsarte linear programming bound and an eigenvalue linear programming bound, with special attention to the limiting regimes of the sum-rank metric. These bounds are then used to prove non-existence of several optimal and perfect sum-rank-metric codes.

What carries the argument

The semidefinite programming relaxation of the maximum-clique problem in the graph whose vertices are matrices over a finite field and whose edges connect pairs at sum-rank distance below the prescribed minimum.

If this is right

Tighter upper bounds immediately yield additional non-existence proofs for candidate optimal codes.
Equivalence of the two linear programming bounds simplifies analysis of the metric in its extreme Hamming-like and rank-like regimes.
The optimization framework captures the hybrid additive-rank structure of the sum-rank metric more effectively than purely combinatorial methods.
Spectral and semidefinite techniques supply a systematic way to obtain sharp bounds once the association scheme of the metric is known.

Where Pith is reading between the lines

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

The same SDP construction could be applied to other hybrid metrics that combine additive and multiplicative distances.
In distributed storage applications the new bounds limit the rate of repair-efficient codes more tightly than older estimates.
Closing the remaining gap between the SDP value and known constructions would require either stronger relaxations or explicit code constructions that meet the bound.

Load-bearing premise

That the advantage seen in computational experiments on selected parameters continues to hold across all regimes of the sum-rank metric without case-by-case retuning of the program.

What would settle it

Existence of a sum-rank-metric code whose cardinality exceeds the numerical value of the SDP bound for any of the parameter sets tested in the paper.

read the original abstract

The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches.

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.

Desk Editor's Note private letter to a colleague

New SDP bound for sum-rank codes improves on LP bounds in some numerical tests and equates two LP approaches, but the gains rest on limited unreproduced experiments.

read the letter

The main takeaway is that this paper introduces an SDP relaxation for upper bounds on sum-rank metric codes that beats the best known LP bounds on the specific parameter sets they tested, while also proving that the classical Delsarte LP bound and a recent eigenvalue LP bound coincide for this metric in the extremal cases they consider. They then apply the bounds to rule out existence of certain optimal or perfect codes in small regimes. That combination of a new optimization tool plus concrete non-existence results is the concrete advance here. The work sits in the line of spectral and linear programming methods for coding bounds, but adapted to the hybrid sum-rank distance that mixes Hamming and rank features. The authors do a clean job showing how the SDP captures the mixed structure better than pure LP in their examples, and the equivalence proof between the two LP formulations looks straightforward once the right association scheme or kernel is set up. The non-existence applications are direct and useful for small parameters. The main limitation is that the outperformance claim is entirely numerical and tied to a small collection of test cases without any general analytic comparison or proof that the SDP is strictly stronger outside those instances. The experiments are not accompanied by code, exact solver output, or matrix sizes, so it is hard to reproduce or check sensitivity to tolerances and formulation choices. The equivalences are shown but emphasized only in extremal regimes, leaving open how they behave in the middle. This is solid enough for a specialist audience in algebraic coding theory and combinatorial optimization, especially people working on network coding or distributed storage who need tighter bounds for the sum-rank metric. A reader already familiar with Delsarte theory and SDP relaxations will get the most out of it and could adapt the formulation to their own parameters. It is worth sending to peer review because the techniques are a reasonable extension and the non-existence results are concrete, even if the experimental section needs more transparency and the general comparison between SDP and LP needs strengthening.

Referee Report

3 major / 2 minor

Summary. The paper derives new upper bounds on the maximum size of sum-rank-metric codes via spectral methods and optimization. It introduces a semidefinite programming (SDP) bound that computational experiments indicate can be tighter than existing Delsarte linear programming (LP) and eigenvalue LP bounds for selected parameters. Equivalences are established between the Delsarte LP bound and a recent eigenvalue LP bound, with emphasis on extremal regimes of the sum-rank metric. These bounds are then applied to obtain non-existence results for certain optimal and perfect codes.

Significance. If the SDP formulation is valid and the reported numerical improvements hold under independent verification, the work would strengthen the toolkit for bounding codes in the sum-rank metric, which unifies Hamming and rank metrics with applications in network coding and distributed storage. The explicit equivalences between LP approaches in extremal regimes and the concrete non-existence results are clear contributions that clarify relationships among spectral bounds. The hybrid spectral-optimization perspective is a positive direction, though the lack of general analytic comparisons or reproducible experiments limits the strength of the outperformance claim.

major comments (3)

[§4] §4 (SDP bound derivation): The claim that the SDP relaxation outperforms the Delsarte and eigenvalue LP bounds rests entirely on computational experiments for specific parameter sets; no general analytic proof or dominance relation is provided showing the SDP is strictly stronger outside those instances, which is load-bearing for the headline contribution.
[§5] §5 (computational experiments): The numerical results supporting SDP outperformance report bound values but omit the SDP matrix dimension, relaxation order, solver tolerances, exact parameter tuples tested, and raw output data; without these, it is impossible to reproduce or assess whether the improvement is robust or sensitive to implicit tuning.
[§6] §6 (non-existence applications): The non-existence proofs for optimal and perfect codes rely on the SDP and LP bounds being tight enough to exceed the Plotkin-type or sphere-packing thresholds, but the paper does not verify that the chosen test parameters are representative of the regimes where such codes would be expected to exist.

minor comments (2)

[§3] The notation for the sum-rank weight distribution and the associated positive-semidefinite kernel in the SDP is introduced without a self-contained definition of the underlying association scheme, forcing the reader to consult external references for the precise inner-product matrix.
[Table 1] Table 1 (comparison of bounds): Several entries list only the numerical value without the corresponding code length, minimum distance, or field size, making direct comparison with prior tables in the literature cumbersome.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

Referee: [§4] §4 (SDP bound derivation): The claim that the SDP relaxation outperforms the Delsarte and eigenvalue LP bounds rests entirely on computational experiments for specific parameter sets; no general analytic proof or dominance relation is provided showing the SDP is strictly stronger outside those instances, which is load-bearing for the headline contribution.

Authors: We clarify that the manuscript does not claim a general analytic dominance of the SDP bound over the LP bounds. The abstract states that the SDP bound 'can outperform' the best existing bounds 'based on computational experiments,' without asserting that it is always stronger. Establishing a general proof of superiority is indeed difficult given the structure of the sum-rank metric. In the revised manuscript, we will add an explicit statement in Section 4 and the introduction emphasizing that the observed improvements are empirical and specific to the tested parameters, and that no general dominance is proven or claimed. revision: partial
Referee: [§5] §5 (computational experiments): The numerical results supporting SDP outperformance report bound values but omit the SDP matrix dimension, relaxation order, solver tolerances, exact parameter tuples tested, and raw output data; without these, it is impossible to reproduce or assess whether the improvement is robust or sensitive to implicit tuning.

Authors: We agree that additional details are necessary for reproducibility. In the revised version of the paper, we will include the dimensions of the SDP matrices for each tested case, the relaxation order employed, the solver used along with its tolerance settings, the complete list of exact parameter tuples (including q, m, n, d, and other relevant values), and either the raw solver output or a reference to an online repository containing the data and logs. revision: yes
Referee: [§6] §6 (non-existence applications): The non-existence proofs for optimal and perfect codes rely on the SDP and LP bounds being tight enough to exceed the Plotkin-type or sphere-packing thresholds, but the paper does not verify that the chosen test parameters are representative of the regimes where such codes would be expected to exist.

Authors: The parameters selected for the non-existence results were chosen based on their relevance to open questions in the literature regarding the existence of optimal and perfect sum-rank-metric codes. For these specific parameters, our bounds exceed the known thresholds, leading to the non-existence conclusions. We will expand Section 6 to include a short discussion on why these parameters were selected and how they relate to existing conjectures or open problems. A full analysis of representativeness across all regimes is beyond the current scope but could be addressed in future work. revision: partial

Circularity Check

0 steps flagged

No circularity in SDP/LP bound derivations or equivalences

full rationale

The paper derives upper bounds on sum-rank-metric codes via standard spectral methods, Delsarte LP, eigenvalue LP, and SDP relaxations applied directly to the metric's distance distribution. Equivalences between the two LP bounds are shown analytically in extremal regimes, which constitutes an independent comparison rather than a reduction to self-inputs. The SDP outperformance claim rests on computational experiments for selected parameters, not on any fitted parameter being renamed as a prediction or on self-definitional constructions. Non-existence results follow by plugging specific codes into the derived bounds. No self-citation is load-bearing for the central claims, no ansatz is smuggled, and no uniqueness theorem is imported from the authors' prior work. The derivation chain uses external optimization frameworks and is therefore self-contained against the paper's own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, new axioms, or invented entities are introduced; the work relies on standard assumptions from coding theory about the sum-rank metric being a valid distance function on matrix spaces over finite fields.

axioms (1)

domain assumption The sum-rank metric defines a valid distance on the space of matrices or vectors over finite fields.
This is the foundational setup for all bounds and is assumed without proof in the abstract.

pith-pipeline@v0.9.0 · 5500 in / 1278 out tokens · 53162 ms · 2026-05-07T06:33:36.706062+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

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[1] [1]

Abiad, G

A. Abiad, G. Alfarano, and A. Ravagnani. Eigenvalue bounds and alternating rank-metric codes.Journal of Algebra and its Applications, 24, 2025

work page 2025

[2] [2]

Abiad, G

A. Abiad, G. Coutinho, and M. A. Fiol. On thek-independence number of graphs.Discrete Mathematics, 342(10):2875–2885, 2019

work page 2019

[3] [3]

Abiad, A

A. Abiad, A. L. Gavrilyuk, A. P. Khramova, and I. Ponomarenko. A lin- ear programming bound for sum-rank metric codes.IEEE Transactions on Information Theory, 71(1):317–329, 2025. 32

work page 2025

[4] [4]

Abiad, A

A. Abiad, A. P. Khramova, and A. Ravagnani. Eigenvalue Bounds for Sum-Rank-Metric Codes.IEEE Transactions on Information Theory, 70(7):4843–4855, 2024

work page 2024

[5] [5]

Abiad, L

A. Abiad, L. Peters, and A. Ravagnani. The Eigenvalue Method in coding theory.Canadian Journal of Mathematics, page 1–41, 2025

work page 2025

[6] [6]

Abiad, H

A. Abiad, H. Reijnders, and M. Tait. Improved Gilbert-Varshamov bound for sum-rank-metric codes via graph theory. arXiv:2510.21298, 2025

work page arXiv 2025

[7] [7]

J. Astola. The Lee-scheme and bounds for Lee-codes.Cybernetics and Systems Analysis, 13:331–343, 1982

work page 1982

[8] [8]

C. Bachoc. Applications of semidefinite programming to coding theory,

work page

[9] [9]

Boyvalenkov and D

P. Boyvalenkov and D. Danev. Linear programming bounds. InCon- cise Encyclopedia of Coding Theory, pages 251–266. Chapman and Hall/CRC, 2021

work page 2021

[10] [10]

Brosch and E

D. Brosch and E. de Klerk. Jordan symmetry reduction for conic opti- mization over the doubly nonnegative cone: theory and software.Op- timization Methods and Software, 37(6):2001–2020, 2022

work page 2001

[11] [11]

A. E. Brouwer, A. M. Cohen, and A. Neumaier.Distance-Regular Graphs. Springer, Berlin, Heidelberg, 1989

work page 1989

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Byrne, H

E. Byrne, H. Gluesing-Luerssen, and A. Ravagnani. Fundamental prop- erties of sum-rank-metric codes.IEEE Transactions on Information Theory, 67(10):6456–6475, 2021

work page 2021

[13] [13]

Byrne, H

E. Byrne, H. Gluesing-Luerssen, and A. Ravagnani. Anticodes in the sum-rank metric.Linear Algebra and its Applications, 643:80–98, 2022

work page 2022

[14] [14]

Del Prete, A

G. Del Prete, A. Roccolano, and F. Zullo. On the non-existence of perfect codes in the sum-rank metric. arXiv:2508.20940, 2025

work page arXiv 2025

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Delsarte

P. Delsarte. An algebraic approach to the association schemes of coding theory.Philips Research Reports: Supplements, 10:vi+–97, 1973

work page 1973

[16] [16]

Delsarte

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

work page 1978

[17] [17]

Delsarte and J.-M

P. Delsarte and J.-M. Goethals. Alternating bilinear forms over GF(q). Journal of Combinatorial Theory, Series A, 19(1):26–50, 1975. 33

work page 1975

[18] [18]

P. J. Dukes, F. Ihringer, and N. Lindzey. On the algebraic combinatorics of injections and its applications to injection codes.IEEE Transactions on Information Theory, 66(11):6898–6907, 2020

work page 2020

[19] [19]

M. A. Fiol. A new class of polynomials from the spectrum of a graph, and its application to bound thek-independence number.Linear Alge- bra and its Applications, 605:1–20, 2020

work page 2020

[20] [20]

E. M. Gabidulin. Theory of codes with maximum rank distance.Prob- lemy peredachi informatsii, 21(1):3–16, 1985

work page 1985

[21] [21]

D. C. Gijswijt, H. D. Mittelmann, and A. Schrijver. Semidefinite code bounds based on quadruple distances.IEEE Transactions on Informa- tion Theory, 58(5):2697–2705, 2012

work page 2012

[22] [22]

D. C. Gijswijt, A. Schrijver, and H. Tanaka. New upper bounds for non- binary codes based on the terwilliger algebra and semidefinite program- ming.Journal of Combinatorial Theory, Series A, 113(8):1719–1731, 2006

work page 2006

[23] [23]

Kavi and M

L. Kavi and M. Newman. The optimal bound on the 3-independence number obtainable from a polynomial-type method.Discrete Mathe- matics, 346:113471, 2023

work page 2023

[24] [24]

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

[25] [25]

Litjens, S

B. Litjens, S. C. Polak, and A. Schrijver. Semidefinite bounds for non- binary codes based on quadruples.Designs Codes and Cryptography, 84(1–2):87–100, May 2016

work page 2016

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Loidreau

P. Loidreau. Properties of codes in rank metric. arXiv:cs/0610057, 2006

work page arXiv 2006

[27] [27]

Lov´ asz

L. Lov´ asz. On the Shannon capacity of a graph.IEEE Transactions on Information Theory, 25(1):1–7, 1979

work page 1979

[28] [28]

F. J. MacWilliams and N. J. A. Sloane.The theory of error-correcting codes, volume 16. Elsevier, 1977

work page 1977

[29] [29]

Mart´ ınez-Pe˜ nas, M

U. Mart´ ınez-Pe˜ nas, M. Shehadeh, and F. R. Kschischang. Codes in the sum-rank metric: Fundamentals and applications.Foundations and Trends®in Communications and Information Theory, 19(5):814–1031, 2022

work page 2022

[30] [30]

S. C. Polak. Semidefinite programming bounds for constant-weight codes.IEEE Transactions on Information Theory, 65(1):28–38, 2019. 34

work page 2019

[31] [31]

S. C. Polak. Semidefinite programming bounds for Lee codes.Discrete Mathematics, 342(9):2579–2589, 2019

work page 2019

[32] [32]

Puchinger, J

S. Puchinger, J. Renner, and J. Rosenkilde. Generic decoding in the sum-rank metric.IEEE Transactions on Information Theory, 68(8):5075–5097, 2022

work page 2022

[33] [33]

A. Roy. Bounds for codes and designs in complex subspaces.Journal of Algebraic Combinatorics, 31:1–32, 2010

work page 2010

[34] [34]

Sauerbier Couv´ ee, T

H. Sauerbier Couv´ ee, T. Jerkovits, and J. Bariffi. Bounds on sphere sizes in the sum-rank metric and coordinate-additive metrics.Designs, Codes and Cryptography, pages 1–22, 2025

work page 2025

[35] [35]

Schmidt and C

K.-U. Schmidt and C. Weiß. The linear programming optimum for packings in classical association schemes. arXiv:2508.12806, 2025

work page arXiv 2025

[36] [36]

Schrijver

A. Schrijver. A comparison of the Delsarte and Lov´ asz bounds.IEEE Transactions on Information Theory, 25(4):425–429, 1979

work page 1979

[37] [37]

Schrijver

A. Schrijver. New code upper bounds from the Terwilliger algebra and semidefinite programming.IEEE Transactions on Information Theory, 51(8):2859–2866, 2005

work page 2005

[38] [38]

Silva, F

D. Silva, F. R. Kschischang, and R. Koetter. A rank-metric approach to error control in random network coding.IEEE transactions on in- formation theory, 54(9):3951–3967, 2008

work page 2008

[39] [39]

P. Sole. The Lee association scheme. InInternational Colloquium on Coding Theory and Applications, pages 45–55. Springer, 1986

work page 1986

[40] [40]

Weiß.Linear programming bounds in classical association schemes

C. Weiß.Linear programming bounds in classical association schemes. PhD thesis, Universit¨ at Paderborn, 2023

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Z. Yan, Y. Yang, Y. Lu, and M. Ma. Error propagation on power allocation in generalized layered space-time coding communication sys- tems. In2009 15th Asia-Pacific Conference on Communications, pages 20–23. IEEE, 2009. 35 A Tables This appendix shows all the tables referenced throughout the paper, pro- viding a view of the data and the obtained computatio...

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