Hamming and simplex codes for the sum-rank metric

Umberto Mart\'inez-Pe\~nas

REVIEW 3 major objections 3 minor 23 references

Hamming and simplex codes for the sum-rank metric

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper defines sum-rank Hamming codes as the longest single-error-correcting codes for a fixed redundancy, and proves that in the one-row case they correspond exactly to maximal-size partial spreads.

desk verdict A genuinely useful new class of sum-rank Hamming codes, with a real false statement in the central m=1 classification that is fixable but must be addressed before the paper is reliable. read the letter →

arxiv 1908.03239 v2 pith:SQSBATFA submitted 2019-08-08 cs.IT math.IT

Umberto Martínez-Peñas This is my paper · ORCID

classification cs.ITmath.IT MSC 94B0594B3594B65

keywords Hammingcodessum-rankmetricpartialspreadsperfectsimplexmultishotnetworkcodinglocallyrepairablesyndromedecoding

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

This paper introduces sum-rank Hamming codes, defined as the longest linear codes with minimum sum-rank distance at least 3, hence single-error-correcting, for a fixed redundancy, base-field size, and field-extension degree. The goal is to generalize classical Hamming codes to the sum-rank metric while maximizing information rate. The main structural result is that when the field-extension degree is 1, sum-rank Hamming codes are exactly the parity-check codes whose column spaces form a maximal-size partial spread, so isometry classes of such codes match maximal partial spreads one-to-one. In that setting, proper sum-rank Hamming codes are perfect when the block length divides the redundancy, and their duals, the sum-rank simplex codes, admit distance bounds derived from spread-size bounds. These codes apply to single-error correction in multishot matrix-multiplicative channels and to binary locally repairable codes.

What carries the argument

The load-bearing machinery is the parity-check characterization of minimum sum-rank distance at least 3 (Proposition 9): for $m=1$ it says that a code with parity-check matrix $H=(H_1,\dots,H_\ell)$ has $d_{\mathrm{SR}}\ge 3$ exactly when the column spaces $H_i=\operatorname{Col}(H_i)\subseteq\mathbb{F}_q^r$ are pairwise disjoint and each has dimension equal to its block length $n_i$. This turns code design into the geometric problem of packing disjoint subspaces, and 'longest code' becomes 'maximal-size partial spread'. The perfection argument then runs on the counting identity $|B_1|=1+\ell(q^N-1)$ together with $\ell=(q^r-1)/(q^N-1)$ when $N\mid r$. The isometry classification (Theorem 2) is what upgrades an existence statement to a bijection between isometry classes of codes and of spreads.

What would settle it

Take $q=2$, $m=1$, with sum-rank partition $n=1+2$ and $r=3$, and search for a linear bijection on $\mathbb{F}_q^n$ that preserves sum-rank weight but sends the 1-dimensional block into the 2-dimensional block. Such a map would violate the omitted step in Theorem 2 and would make Theorem 6's bijection between improper Hamming codes and improper partial spreads fail; a computer search over small parameters can settle this directly.

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Extended reading notes

Core claim

The paper's central claim is that, for $m=1$, a linear code over $\mathbb{F}_q$ is a proper sum-rank Hamming code with redundancy $r$ and equal sublengths $N$ if and only if it admits a parity-check matrix $H=(H_1,\dots,H_\ell)$ whose column spaces $H_i=\operatorname{Col}(H_i)\subseteq\mathbb{F}_q^r$ form a maximal-size partial $N$-spread (Theorem 3). Consequently, sum-rank isometry classes of proper sum-rank Hamming codes are in bijection with maximal-size partial $N$-spreads in $\mathbb{F}_q^r$ (Theorem 4). When $N$ divides $r$, the number of blocks reaches $(q^r-1)/(q^N-1)$ and the codes are perfect for the sum-rank metric: the balls of radius 1 around codewords tile the entire space (Corollary 15). The same dictionary extends to unequal sublengths, now using maximal-size improper partial spreads, provided the linear isometry classification from Theorem 2 holds; the paper leaves that proof to the reader.

Load-bearing premise

The classification relies on the unproved claim that in a linear sum-rank isometry with unequal block sizes, a block of one size cannot be mapped to a block of a different size; if that claim fails, the correspondence between improper sum-rank Hamming codes and improper partial spreads collapses.

Editorial extensions

If this is right

For $m=1$ and equal sublengths $N$, the length of a proper sum-rank Hamming code with redundancy $r$ is bounded between the two expressions in Corollary 13, with equality when $N$ divides $r$.
When $N\mid r$, proper sum-rank Hamming codes are perfect for the sum-rank metric, so the single-error-correcting capability is exactly 1 and every error of sum-rank weight at most 1 is uniquely correctable.
The syndrome decoding algorithm for $m=1$ finds the error location and value by solving linear systems against the partial-spread subspaces, with complexity $O(nr+\ell r^3)$ over $\mathbb{F}_q$.
The dual sum-rank simplex codes have minimum sum-rank distance at least the lower bound in Theorem 7, expressed in terms of the remainder of $r$ modulo $N$.
The codes correct a single sum-rank error in multishot matrix-multiplicative channels, and for $q=2$, $m=1$ they provide locally repairable codes with locality $N$ that can correct any pattern with one erasure per local group plus two additional erasures anywhere.

Reading between the lines

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

Inference: Because the correspondence with partial spreads is bijective, any future improvement in bounds on maximal partial spreads automatically improves the length bounds in Corollary 13 and the distance bound for sum-rank simplex codes; the paper itself uses only the simple bounds from [4] and [10].
Inference: For $m>1$, the same recipe might yield matrix-space analogues of sum-rank Hamming codes by replacing disjoint subspaces with pairwise rank-disjoint matrix spaces; the paper leaves the existence of nontrivial perfect sum-rank codes open in that regime.
Inference: The binary locally repairable code construction of Theorem 9 is flexible enough to be combined with arbitrary local linear codes, which could produce hierarchical locally repairable codes even though the paper only notes the possibility in passing.

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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

Short version: this paper has a real idea—define Hamming codes for the sum-rank metric as the longest single-error-correcting codes, and tie the m=1 case to maximal partial spreads—and the N|r case is correct. But the main classification theorem is false as stated, and there is a false lemma for m>1. Worth reading, worth fixing, not acceptable as is.

What is new: sum-rank Hamming and simplex codes, the bijection with partial spreads (when it holds), the observation that for N|r they are perfect, the simple syndrome decoder, and the binary LRC construction. The perfect-code result (Corollary 15) is clean and the counting is right; the decoder in Section 3.4 is correct and practical; the LRC application in Theorem 9 is a nice use of binary sum-rank Hamming codes.

Soft spots, roughly in increasing order of severity.

First, Table 1's first row is wrong: for N=2, r=4, Theorem 9 gives ℓ=5, M=15, k=6, not ℓ=4, M=12. Minor typo, but it should be fixed.

Second, the abstract and introduction claim sum-rank Hamming codes are perfect for m=1 without stating the N|r condition; Corollary 15 only holds when N divides r. Overstatement.

Third, several proofs are omitted. Theorem 2's extension to unequal sublengths is "left to the reader", and Theorems 5 and 6 are also left to the reader. These are not trivial; the improper isometry classification depends on the missing argument.

Fourth, Proposition 9 is stated for all m and is false for m>1. For q=2, m=2, take H_i = [1,0,ω; 0,1,ω^2] in F_4^{2×3}. Conditions 1 and 2 hold, but c=(ω,ω^2,1) has rank 2 and zero syndrome. So the characterization of d_SR≥3 via the two syndrome conditions is only valid for m=1. The paper only uses it for m=1, but the statement as written is wrong.

Fifth, the main issue: Theorems 3 and 4 claim a bijection between proper sum-rank Hamming codes of redundancy r and maximal-size partial N-spreads in F_q^r, but no spanning condition is imposed. The stress-test example is right: for q=2, r=3, N=2, a single plane is a maximal-size partial 2-spread, but no proper sum-rank Hamming code with redundancy 3 exists, because a parity-check matrix built from one 2D block has rank 2. The correct statement needs the partial spread to span F_q^r, which is automatic when N divides r, so Corollaries 13 and 15 survive. But the theorem as written is false.

Bottom line: the N|r case and the LRC application are sound and worth citing. The paper deserves a serious referee—the idea is good and the flaws are fixable—but the current version should not be taken as reliable, especially the general proper-case classification and Proposition 9. Send it to review, ask for the spanning hypothesis, the missing proofs, and a correction to Proposition 9.

Referee Report

3 major / 3 minor

Summary. The paper introduces sum-rank Hamming codes, defined as the longest linear codes with minimum sum-rank distance at least 3 for a fixed redundancy, base field, and extension degree, together with their duals, sum-rank simplex codes. General bounds on code length and number of shots are given. For extension degree m=1, the paper characterizes proper sum-rank Hamming codes via maximal-size partial spreads, proves they are perfect when the sublength divides the redundancy, provides a syndrome decoding algorithm, and derives bounds for simplex codes. Applications to multishot matrix-multiplicative channels and binary locally repairable codes are also presented.

Significance. If the m=1 classification were correct as stated, it would be an elegant and useful bridge between single-error sum-rank correction and finite-geometry partial spreads, with practical benefits including an efficient decoding algorithm and explicit binary LRC constructions. The perfect-code result for N dividing r is a natural sum-rank analogue of the classical Hamming-code perfectness. However, the central bijection is false without an additional spanning condition, and several load-bearing proofs for the unequal-sublength (improper) case are omitted. The core idea is salvageable, but the manuscript needs substantive revision before the claims are supported.

major comments (3)

[Section 2, Theorem 2 and Section 3.3, Theorems 5 and 6] The stated equivalence between proper sum-rank Hamming codes and maximal-size partial N-spreads is false when a maximal-size partial N-spread does not span F_q^r. For q=2, r=3, N=2, a maximal-size partial 2-spread in F_2^3 has size 1 (any two distinct 2-dimensional subspaces intersect nontrivially), so it exists. But no proper sum-rank Hamming code with these parameters exists: since m=1, any nonzero codeword has sum-rank weight at most the number of blocks, so d_SR(C)>=3 forces at least 3 blocks; meanwhile a parity-check matrix with redundancy r=3 would need three pairwise disjoint 2-dimensional column spaces in F_2^3, which is impossible. Equivalently, a parity-check matrix assembled from a single 2-dimensional column space can have rank at most 2, not the required rank 3. Thus the right-hand side of Theorem 3 can be nonempty while the left-hand side is empty, and the bijection in Theorem 4 fails in one direction. The statements need a spanning condition on the partial spread, or an explicit hypothesis that the union of the subspaces in the spread equals F_q^r, and the existence of such spreads must be checked for the relevant parameters. This issue affects Corollary 13, Corollary 15, and Theorem 7, which all rely on Theorem 3.
[Section 3.3, Proposition 9] The proof of Theorem 2 in the unequal-sublength case is omitted with the sentence 'left to the reader', but this case is exactly what is needed for the improper sum-rank Hamming code classification. Theorems 5 and 6 then state the improper analogue of the spread correspondence with 'Proofs are left to the reader' as well. Since the improper-case classification is an advertised contribution and is not reduced to previously proved results, these are load-bearing gaps. The authors should either provide complete proofs or explicitly restrict the paper's claims to the proper case until the unequal-sublength isometry classification is established.
[Section 5.2, Table 1] Proposition 9 is the key characterization of codes with minimum sum-rank distance at least 3 in the m=1 case, and it is used directly in the proof of Theorem 3, yet its proof is only asserted to be 'immediate'. The statement is plausible and likely correct, but because the entire partial-spread correspondence rests on it, a full proof (or at least a detailed derivation from Theorem 1 and Lemma 6) should be included so the reader can verify both conditions and their necessity.

minor comments (3)

[Section 3.3, Notation] The first row of Table 1 is internally inconsistent with the formulas in Theorem 9. For q=2, N=2, r=4, the formulas give ℓ=(2^4-1)/(2^2-1)=5, M=(N+1)ℓ=15, and k=Nℓ-r=6. The table lists ℓ=4, M=12, and k=6; the first two entries should be corrected to 5 and 15, respectively.
[Section 3.4, Complexity remark] In Theorems 3 and 4, the same symbol H_i denotes both a matrix block and its column space (with the latter written as H_i = Col(H_i)). This overloading makes the statement of the spread conditions harder to read; a separate notation, such as overlined or script letters for subspaces, would improve clarity.
[Section 1, Abstract] The final complexity sentence says the term (q^r-1)/(q^N-1) is dominant over r^3+Nr and hence the running time is 'close to linear in the code length n'. This is true for fixed q and N as r grows, but it may be helpful to state the asymptotic regime explicitly, since for small r the polynomial term can be comparable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core reductions go to external partial-spread theory and independent prior theorems, not back to the paper's own conclusions.

full rationale

The derivation chain is not circular. Sum-rank Hamming codes are defined by an extremal property (longest code with d_SR >= 3 for fixed redundancy r), and the main m=1 results relate that extremal property to maximal-size partial spreads. The paper's load-bearing inputs are external and independent: the partial-spread bounds in Proposition 12 come from Beutelspacher and Gorla-Ravagnani, the rank-isometry result in the proof of Theorem 2 comes from Berger, and Theorem 1 is cited from the author's prior work [20]. None of these inputs already contains the target conclusion: Theorem 1 is a distance characterization of arbitrary sum-rank codes, not a statement about Hamming codes or partial spreads, and the perfect-code count in Corollary 15 is an exact parameter-free equality |C|*|B_1| = |F_q^n| with no fitted constants. The self-citations to [18] and [20] are used as lemmas with independent published proofs, not as re-derivations of the Hamming-code/partial-spread classification, so they do not raise the circularity score. Separately, there are mathematical soundness gaps that are not circularity: the unequal-sublength case of Theorem 2 is asserted with the key claim 'left to the reader', and Theorem 3's bijection does not explicitly enforce the full-row-rank condition rank(H)=r, which is needed for a partial spread's column spaces to realize a code with the prescribed redundancy. Those issues are correctness concerns, not input-output equivalence, so the appropriate circularity finding is 'none'.

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

The central results rely on three imported results from prior literature (the sum-rank-to-Hamming characterization, the [20] Singleton-type bound, and partial spread size bounds) plus standard finite-field algebra. No parameters are fitted to data; q, m, r, and N are problem inputs.

assumptions (4)

domain assumption Theorem 1 of [20]: d_SR(C) = min_{A=diag(A_i)} d_H(CA) over invertible F_q block-diagonal matrices
Imported from prior work; used to reduce sum-rank distance to Hamming distance in Proposition 7 and Proposition 9.
domain assumption [20, Cor. 3] Singleton-type bound: |C| <= (q^{n/ell})^(ell m - d_SR(C) + 1)
Used in Proposition 7 to derive n <= ell m r / 2.
domain assumption Partial spread size bounds in Proposition 12, from Beutelspacher and Gorla-Ravagnani: (q^r-q^s)/(q^N-1) - q^s + 1 <= |I| <= (q^r-q^s)/(q^N-1)
External finite-geometry results used for length estimates (Corollary 13) and for the simplex distance bound (Theorem 7).
standard math Standard algebraic facts over finite fields and F_q-linear codes
Rank-nullity, duality, projective space counts, and basis extension for parity-check matrices are used throughout.

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Pith. "Pith review of Hamming and simplex codes for the sum-rank metric." pith.science (2026). https://pith.science/paper/SQSBATFA

@misc{pith2026190803239,
  author       = {Pith},
  title        = {Pith review of: Hamming and simplex codes for the sum-rank metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQSBATFA}},
  note         = {Machine review of arXiv:1908.03239}
}

abstract

Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least $ 3 $ (thus one-error-correcting) for a fixed redundancy $ r $, base-field size $ q $ and field-extension degree $ m $ (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is $ 1 $, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads. In that case, it is also shown that sum-rank Hamming codes are perfect codes for the sum-rank metric. Also in that case, estimates on the parameters (lengths and number of shots) of sum-rank Hamming codes are given, together with an efficient syndrome decoding algorithm. Duals of sum-rank Hamming codes, called sum-rank simplex codes, are then introduced. Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads. As applications, sum-rank Hamming codes are proposed for error correction in multishot matrix-multiplicative channels and to construct locally repairable codes over small fields, including binary.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Barra and H

A. Barra and H. Gluesing-Luerssen. MacWilliams extensi on theorems and the local-global property for codes over Frobenius rings. J. Pure Appl. Algebra , 219(4):703–728, 2015

work page 2015

[2] [2]

T. P. Berger. Isometries for rank distance and permutati on group of Gabidulin codes. IEEE Trans. Inform. Theory , 49(11):3016–3019, 2003

work page 2003

[3] [3]

Beutelspacher

A. Beutelspacher. Partial spreads in ﬁnite projective s paces and partial designs. Mathematische Zeitschrift, 145(3):211–229, Oct 1975

work page 1975

[4] [4]

Beutelspacher

A. Beutelspacher. On t-covers in ﬁnite projective spaces. Journal of Geometry , 12(1):10–16, 1979

work page 1979

[5] [5]

Delsarte

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

work page 1978

[6] [6]

T. Etzion. Perfect byte-correcting codes. IEEE Trans. Info. Theory , 44(7):3140–3146, Nov 1998

work page 1998

[7] [7]

Etzion and G

T. Etzion and G. Greenberg. Constructions for perfect mi xed codes and other covering codes. IEEE Trans. Info. Theory , 39(1):209–214, 1993

work page 1993

[8] [8]

E. M. Gabidulin. Theory of codes with maximum rank distan ce. Problems of Information Trans- mission, 21(1):1–12, 1985

work page 1985