On the non-existence of perfect codes in the sum-rank metric

Antonio Roccolano; Ferdinando Zullo; Giuseppe Del Prete

REVIEW 2 major objections 5 minor 1 cited by

On the non-existence of perfect codes in the sum-rank metric

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that perfect codes in the sum-rank metric, codes whose metric balls tile the whole space exactly, can exist in the two-block square-matrix case only for small fields and small radii, and rules out many multi-block parameter

desk verdict Solid extension of Loidreau's rank-metric argument to sum-rank perfect codes, with several correct non-existence results, but the headline two-block parameter exclusion has a real gap for radii k=2,3. read the letter →

arxiv 2508.20940 v1 pith:IGY476KV submitted 2025-08-28 cs.IT math.COmath.IT

Giuseppe Del Prete , Antonio Roccolano , Ferdinando Zullo This is my paper · ORCID

classification cs.ITmath.COmath.IT MSC 11T7151E2094B05

keywords sum-rankmetricperfectcodessphere-packingboundSingleton-likerank-metricfinitefieldsballvolumesnon-existence

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 tries to show that perfect codes in the sum-rank metric—codes whose disjoint metric balls partition the entire ambient space—are extremely rare, and to narrow the open question of which parameter sets can host them. In the two-block space F_q^{n×n} ⊕ F_q^{n×n}, it proves that any perfect code must have either q in {2,3,4} or 5 ≤ q < e^3 with radius at most 5, and it eliminates the q=2 and q=3 cases for many block sizes. For spaces with more than two blocks, it proves nonexistence under several distance, divisibility, and dimension conditions, and adds exact congruence checks as computational evidence. A reader should care because the sum-rank metric models multishot network and space-time coding, and knowing which parameters admit perfect codes settles the limits of error-correction by tiling.

What carries the argument

The load-bearing identity is the sphere-packing equality |C| V_k = |Mat|, which a perfect code satisfies by exactly tiling the space with disjoint radius-k balls. This is coupled with the Singleton-like bound |C| ≤ q^{...} to turn perfection into a lower bound on ball volume, q^{2kn} ≤ V_k. The volume is bounded above by counting rank-k matrix spheres in t copies of F_q^{n×n}: V_k ≤ k(k+1) binom(k+t-1,t-1) q^{(2n+1-k/t)k + (4-t)/4}, where the binomial factor controls the number of ways to split radius k among t blocks. For t=2 the two bounds combine into q^{(k^2-2k-1)/2} ≤ k(k+1)^2, whose failure for large q or k rules out perfect codes. A second mechanism is the congruence V_k ≡ 1 + Σ_{i=1}

What would settle it

Exhaustively enumerate linear codes in F_5^{2×2}⊕F_5^{2×2} with minimum distance 13 (decoding radius k=6) and check whether any achieves |C|·V_6 = 5^8; a hit would directly contradict Theorem 4.1, and a complete enumeration with no hit would support it. A second check: compute the exact volume V_6(Mat) for that space and compare it with the Proposition 3.8 upper bound; if the exact volume exceeds the bound, the proof's necessary-condition step collapses.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the sphere-packing equality |C| V_k = q^{2n^2} in F_q^{n×n}⊕F_q^{n×n} is incompatible with the Singleton-like bound except in a narrow parameter window: combining them forces q^{(k^2-2k-1)/2} ≤ k(k+1)^2, and this single inequality rules out q ≥ 5 with k ≥ 6 and q > e^3 with k ≤ 5. The same comparison, with enlarged volume bounds, yields non-existence for many t-block spaces when the minimum distance is large relative to t, while congruence computations for ball volumes modulo the field characteristic rule out additional (t,k) pairs for all extension fields of characteristics 2, 3, 5, and 7. The authors do not claim to have found any per

Load-bearing premise

The paper's large-distance exclusions all rest on an upper bound for the size of a metric ball that the authors themselves say is not tight; if that bound is too loose in the regimes where it is used, the contradictions do not go through.

Editorial extensions

If this is right

In the two-block square case, any future search for perfect sum-rank codes can be restricted to q∈{2,3,4} or 5≤q<e^3 with k=floor((d-1)/2)≤5; no code exists outside this window.
For q=2, when n≥4 the only possible radius left is k=2, and for q=3 all radii are excluded when n≥3.
With more than two blocks, perfect codes of radius 1 require q≤t+1, and perfect codes of minimum distance 5 or 6 require t≡1 or 2 mod q for odd q, plus code dimension divisible by n.
Whenever gcd(q,k!)=1, any perfect code forces gcd(t,q)=1, so t sharing a prime with q is ruled out for those radii.
The congruence-based tables exclude perfect codes for many (t,k) pairs in every extension of F_2, F_3, F_5, and F_7, independent of the extension degree.

Reading between the lines

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

The same volume-Singleton comparison should apply to non-square block spaces or mixed block sizes, yielding analogous but likely different parameter windows; the paper does not work those cases out.
The remaining window—q∈{2,3,4} or 5≤q<e^3 with k≤5—is small enough that a targeted computer search for codes other than the Hamming-derived ones becomes a realistic next step.
The congruence lemma V_k ≡ 1+Σ(-1)^i binom(t,i) mod q is a cheap sieve: it can be evaluated exactly for large k without computing full ball volumes, so the tables for characteristics p≥11 could be extended by the same method.
If the nonexistence pattern holds, the sum-rank perfect-code landscape would mirror the Hamming classification: the only genuine perfect codes would be trivial or Hamming-induced, closing the question opened by earlier work.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper genuinely extends Loidreau's rank-metric non-existence technique to the sum-rank metric, and most of the machinery is sound. The volume bounds in Section 3, the explicit small-radius ball computations, and the congruential arguments in Section 5 (Lemma 5.7, Theorems 5.8, Proposition 5.10) are careful and give real new constraints. The multi-block non-existence results, especially the sphere-packing plus Singleton bound comparisons, appear to work. This is a useful contribution to the sum-rank coding subfield and answers part of Martinez-Peñas's question.

But the paper's own headline case, the two-block space F_q^{n x n} ⊕ F_q^{n x n}, has a gap that the reader's stress-test note correctly identifies. Theorem 4.2 claims to rule out radii k ≤ 5 for q > e^3, but the proof reduces to inequality (13), q^{(k^2-2k-1)/2} ≤ k(k+1)^2. For k=2 this is q^{-1/2} ≤ 18, which holds for every q. For k=3 it gives q ≤ 48, which does not contradict q > e^3 ≈ 20.09. So the proof as written leaves k=2,3 open. Corollary 4.3, which is advertised in the abstract, is not established by the given argument. The authors do handle small k for q=2,3 in the appendices, but not for general q in the range 5 ≤ q < e^3 or for q in (e^3,48]. This is a real gap, not just a loose bound.

I also want to flag that Proposition 3.8's upper bound is admittedly not tight (the paper says so at the end of Section 3). Since several of the large-distance contradictions hinge on that bound, the constants matter. The multi-block theorems may survive, but each use of that bound should be checked.

On the positive side, the paper is clearly written, the citations are appropriate, and the computational tables in Section 6 are reproducible from the congruence criterion. The self-citations are background and not a problem.

Who is this for? Researchers working on sum-rank metric codes and covering problems. It deserves a serious referee, but with the expectation that the two-block case needs fixing: either prove the k=2,3 exclusions directly or weaken Corollary 4.3 accordingly. I would not cite the two-block constraint as stated until that is resolved.

Referee Report

2 major / 5 minor

Summary. The paper studies linear perfect codes in the sum-rank metric, i.e. codes attaining equality in the sphere-packing bound for spaces of the form Mat(n,m,F_q) = F_q^{n_1×m_1} ⊕ ... ⊕ F_q^{n_t×m_t}. It derives lower and upper bounds on sphere/ball volumes (Propositions 3.7 and 3.8), then combines the sphere-packing bound with the Singleton-like bound to obtain non-existence results. The main focus is the two-block case F_q^{n×n} ⊕ F_q^{n×n}: Theorem 4.1 excludes radii k ≥ 6 when q ≥ 5, Theorem 4.2 aims to exclude k ≤ 5 when q > e^3, and Corollary 4.3 consequently restricts possible parameters to q ∈ {2,3,4} or 5 ≤ q < e^3 with k ≤ 5. For t > 2 blocks, the paper proves several non-existence statements depending on q, t, parity of the distance, and divisibility conditions (Propositions 5.1, 5.2, 5.4, Theorems 5.5, 5.6, 5.8, Proposition 5.10), and provides computational tables based on congruence conditions on ball volumes. An appendix handles the q=2 and q=3 two-block cases for small radii. The central two-block result is only partially established: Theorem 4.2 does not actually rule out k=2 and k=3 for q>e^3, so Corollary 4.3 is not proven from the given argument.

Significance. If fully established, the paper would give the first systematic non-existence results for perfect codes in the sum-rank metric beyond the rank-metric case, considerably narrowing the parameter space left open by Martínez-Peñas. The approach is transparent and uses no fitted parameters: the derivations are explicit and the volume formulas for small balls are concrete. The paper also correctly identifies that its upper bound in Proposition 3.8 is not tight, and the use of this bound as an upper bound in necessary inequalities is methodologically sound even when non-tight. However, the current main theorem for two blocks has a specific gap, and one multi-block theorem is stated without a proof. The announced parameter constraints are therefore conditional, and the broad title/abstract overstate what is proved.

major comments (2)

[Theorem 4.2 and Corollary 4.3 (Section 4, inequality (13))] The proof of Theorem 4.2 claims that inequality (13), q^{(k^2-2k-1)/2} ≤ k(k+1)^2, contradicts q>e^3 for every k≤5. This is false for k=2 and k=3. For k=2, (13) reads q^{-1/2} ≤ 18, which holds for all q. For k=3, it reads q ≤ 48, which is compatible with q>e^3 (e.g. q=27,32,37,41,43,47). The later step in the proof uses the factor k^2-11, which is positive only for k≥4; for k=3 it is negative, so the direction of the inequality is invalid. Thus the argument only excludes k=4,5 from the range k≤5. Corollary 4.3, which is the paper's central two-block conclusion, therefore does not follow for k=2,3. These small radii need a separate treatment (e.g. using the exact formulas in Proposition 3.4 or the sphere-packing equality directly).
[Theorem 5.6 (Section 5)] Theorem 5.6 is stated without proof, with the text saying only 'With a very similar argument one can show the following.' No derivation is given for the even-distance case; in particular, the analogue of the Singleton-bound step used for odd d (Eq. (11)–(12) in Theorem 4.1) and the way it combines with Proposition 3.8 are not shown. Since Theorem 5.6 is a central non-existence result for the multi-block setting, the proof should be supplied or at least a detailed sketch of the differences from Theorem 5.5.

minor comments (5)

[End of Section 3] The authors note that the bounds in Proposition 3.8 are not tight. Since these bounds are used only as upper bounds in necessary inequalities, non-tightness does not by itself affect the validity of the contradictions; a short remark to this effect would prevent reader confusion.
[Proposition 5.2] The first bullet is logically malformed: 'neither t≢1 (mod q) nor t≢2 (mod q)' should be 't not congruent to 1 or 2 modulo q' (i.e. t≢1 and t≢2 mod q). As written, the double negative gives an unsatisfiable condition.
[Theorem 9.2, Case 1] In the displayed formula for |C|, the number '22n2' appears; it should be '32n2' since q=3 in this appendix. Compare with the correct expression in the surrounding text.
[Section 6] The computational tables are said to be obtained from the congruence criterion, but no script or pseudo-code is provided. For reproducibility, include the algorithm or a reference to where the computation is performed.
[Throughout] There are minor inconsistencies in notation, e.g. Mat(n,m,F_q) is used both for the tuple (n,m) and for the direct-sum space; the authors should make the dependence on the tuples explicit in one place.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained, with only background self-citations.

full rationale

The paper's central non-existence results are obtained by combining external results — the Singleton-like bound [7, Theorem 2.3], Loidreau's rank-metric volume bounds [13, Proposition 3.6], and the sphere-packing bound — with volume estimates that the paper proves by elementary counting (Propositions 3.3, 3.4, 3.7, 3.8). There are no fitted parameters, no data-derived constants, and no 'prediction' that is secretly an input. The self-citations ([18] Mushrraf–Zullo and [29] Zullo) appear only in the introduction as background on restricted-metric covering codes and are not used in any proof. The paper's own admission that the upper bound of Proposition 3.8 is not tight, and the possible failure of Theorem 4.2 to exclude radii k=2,3 for q>e^3, are correctness concerns about the strength of the derived inequalities, not circularity: inequality (13) is genuinely derived from the sphere-packing and Singleton bounds, and the gap does not consist in assuming the conclusion. Thus the derivation chain does not reduce to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central non-existence proofs rely on the cited Singleton-like bound and rank-metric volume bounds, and on the paper's own non-tight sum-rank volume estimates. No free parameters are fitted and no new entities are introduced.

assumptions (4)

domain assumption Volume formula for sum-rank spheres (Prop 3.3) from [7, Section III]
The paper inherits this exact counting formula for sets of matrices of fixed sum-rank weight; all volume computations and congruence results rest on it. Cited to [7], not re-derived.
domain assumption Singleton-like bound for sum-rank codes (Theorem 2.3) from [7, Theorem 3.2]
Used to bound |C| in every non-existence proof; if the bound were weaker, the derived inequalities q^{2kn} <= V_k would not follow.
domain assumption Rank-metric ball and sphere bounds (Prop 3.6) from [13, Prop 1]
Used as input to derive the sum-rank bounds in Props 3.7 and 3.8; the constants in the exponents affect the final thresholds.
standard math Restricted partition count bound |tau_{k,t,n}| <= C(k+t-1,t-1) from [21]
Used in Prop 3.8 to bound the number of block rank tuples; if the count were larger, the upper bound would need an extra factor.

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Pith. "Pith review of On the non-existence of perfect codes in the sum-rank metric." pith.science (2026). https://pith.science/paper/IGY476KV

@misc{pith2026250820940,
  author       = {Pith},
  title        = {Pith review of: On the non-existence of perfect codes in the sum-rank metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGY476KV}},
  note         = {Machine review of arXiv:2508.20940}
}

read the original abstract

We study perfect codes in the sum-rank metric, a generalization of both the Hamming and rank metrics relevant in multishot network coding and space-time coding. A perfect code attains equality in the sphere-packing bound, corresponding to a partition of the ambient space into disjoint metric balls. While perfect codes in the Hamming and rank metrics are completely classified, the existence of nontrivial perfect codes in the sum-rank metric remains largely open. In this paper, we investigate linear perfect codes in the sum-rank metric. We analyze the geometry of balls and derive bounds on their volumes, showing how the sphere-packing bound applies. For two-block spaces, we determine explicit parameter constraints for the existence of perfect codes. For multiple-block spaces, we establish non-existence results for various ranges of minimum distance, divisibility conditions, and code dimensions. We further provide computational evidence based on congruence conditions imposed by the volume of metric balls.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Perfect codes in weakly metric association schemes
math.CO 2026-01 reject novelty 5.0 of 10

Perfect codes in Lee, NRT, mixed Hamming, and sum-rank metrics are claimed to be asymptotically nonexistent beyond certain radius/length ranges via a Schwartz-Zippel bound on Lloyd polynomials.

Reference graph

Works this paper leans on

29 extracted references · 27 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Abiad, G

A. Abiad, G. N. Alfarano, and A. Ravagnani. Eigenvalue bounds and alternating rank-metric codes. Journal of Algebra and Its Applications , page 2541011, 2025

work page 2025

[2] [2]

R. Alter. On the non-existence of perfect double Hamming-error-correcting codes on q = 8 and q = 9 symbols. Information and Control , 13(6):619–627, 1968

work page 1968

[3] [3]

R. Alter. On the nonexistence of close-packed double Hamming-error-correcting codes on q = 7 symbols. Journal of Computer and System Sciences , 2(2):169–176, 1968

work page 1968

[4] [4]

Bartoli, M

D. Bartoli, M. Borello, and G. Marino. Saturating linear sets of minimal rank. Finite Fields and Their Applications, 95:102390, 2024

work page 2024

[5] [5]

Bonini, M

M. Bonini, M. Borello, and E. Byrne. Saturating systems and the rank-metric covering radius. Journal of Algebraic Combinatorics, 58(4):1173–1202, 2023

work page 2023

[6] [6]

Bonini, M

M. Bonini, M. Borello, and E. Byrne. The geometry of covering codes in the sum–rank metric. Designs, Codes and Cryptography, pages 1–17, 2025

work page 2025

[7] [7]

Byrne, H

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

work page 2021

[8] [8]

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