Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration

Chaofeng Guan; Gaojun Luo; Hong Wang; Shitao Li; Zhi Ma

REVIEW 4 major objections 5 minor 41 references

Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper establishes iterative lower bounds on the number of minimum-weight codewords in Griesmer-optimal linear codes, and proves these bounds are tight for binary codes up to dimension 5 in most cases, with the remaining gap at most 2.

desk verdict Iterative bounds are a real contribution, but the 5D completeness claims rest on black-box enumerations. read the letter →

arxiv 2507.05567 v1 pith:AGVC334H submitted 2025-07-08 cs.IT math.IT

Chaofeng Guan , Shitao Li , Gaojun Luo , Zhi Ma , Hong Wang This is my paper · ORCID

classification cs.ITmath.IT MSC 94B6594B05

keywords errorcoefficientminimum-weightcodewordsAFER-optimallinearcodesGriesmerbounditerativelowerboundsresidualminihypersbinary

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 is trying to pin down the error coefficient—the number of minimum-weight codewords—of linear codes that attain the Griesmer bound, the classical lower limit on length for a given dimension and minimum distance. Such counts matter because, on an additive white Gaussian noise channel under maximum-likelihood decoding, the smallest error coefficient among codes with the best possible distance gives the lowest asymptotic frame error rate (AFER). The paper proposes five iterative lower bounds on these coefficients, each reducing the problem to smaller-dimensional codes through residual codes, subcode chains, and projective-geometry minihypers, and packages them as a single maximum bound in Theorem 9. For binary codes of dimension at most 5 the bound is tight in most cases; the remaining 5-dimensional AFER-optimal codes are explicitly constructed, and the gap between bound and true value is at most 2. If the claims hold, a practical certificate now exists for deciding which optimal binary codes have the best asymptotic error performance.

What carries the argument

The mechanism is recursion through residual codes and subcodes. For a minimum-weight codeword $c$, the residual code $\Upsilon_c(C)$ has length $n-d$, dimension $k-1$, and minimum distance at least $\lceil d/q\rceil$, so lower bounds for dimension $k-1$ feed into $k$-dimensional bounds; Theorem 4 quantifies this via the count $e_{\lceil d/q\rceil}(n-d,k-1,q)$. Proposition 2 supplies the structural ingredient: a Griesmer-optimal code contains a chain $C_1\subset\cdots\subset C_{k_1}\subset C$ whose effective lengths equal the Griesmer values $g_q(i,d)$, which Theorem 7 exploits through the difference $\varsigma_q(k_2,d)$ between consecutive Griesmer-optimal error coefficients. For codes that are exactly Griesmer, the associated minihyper $\mathfrak{D}(C)$ in projective geometry carries the counting problem, and Theorem 8 uses its rank to multiply a lower-dimensional error coefficient by $$q^{{k-k'}}$$. A separate tool, Lemma 8, characterizes non-extendable even-distance binary codes by the residual parameters $[n-d,k-1,d/2]$, powering the lower bound in Theorem 10 that handles the exceptional 5-dimensional cases.

What would settle it

Exhaustively enumerate all binary Griesmer-optimal $[31s_2+7,5,16s_2+2]_2$ codes for $s_2=0,1,2$ and check whether any has an error coefficient below 5; a single code with $A_d=4$ would refute Theorem 11. A broader test is to search all $[n,5,d]$ Griesmer-optimal codes at the lengths in Table IV and look for an error coefficient more than 2 below the Theorem 9 bound, which would refute the claimed worst-case gap.

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Extended reading notes

Core claim

The central claim is that the error coefficient $A_d(C)$ of a Griesmer-optimal $[n,k,d]_q$ code admits a computable lower bound obtained recursively from lower dimensions. Theorem 9 assembles five bounds $$L_q^{{(1)}}$(n,k),\dots,$L_q^{{(5)}}$(n,k)$ that apply under different hypotheses—unconditionally, or when the code is Griesmer-optimal with certain values of the parameter $\Gamma_q(n,k,d)$, or when the rank of the associated minihyper is known. Iterating these bounds from the two-dimensional AFER-optimal codes of Theorem 3 fixes the error coefficients of all binary AFER-optimal codes up to dimension 4 and most of dimension 5, with explicit generator matrices or point multisets. The paper proves in Theorem 11 that $[31s_2+7,5,16s_2+2;5]_2$ is AFER-optimal and in Theorem 13 that $[31s_1+8,5,16s_1+3;13]_2$, $[31s_1+12,5,16s_1+5;11]_2$, and $[31s_2+20,5,16s_2+9;8]_2$ are AFER-optimal, completing the dimension-5 list through the two classified families of Griesmer point configurations (SS-type and Belov-type minihypers).

Load-bearing premise

The iteration depends on Proposition 2's assertion that every Griesmer-optimal code contains a nested subcode chain with effective lengths exactly $g_q(i,d)$; the proof finds these subcodes by a non-constructive existence argument that assumes a punctured Griesmer-violating code always contains a codeword attaining the minimum in equation (25).

Editorial extensions

If this is right

Every binary Griesmer-optimal linear code of dimension at most 5 can now be certified for AFER-optimality, with an explicit construction and a known error coefficient.
The iteration can be continued: starting from the dimension-2, 3, and 5 databases, Theorem 9 produces lower bounds for 6-dimensional Griesmer-optimal codes and, in principle, for all higher dimensions.
Codes constructed in earlier work whose AFER-optimality could not be decided by the linear programming bound are settled by the new bounds, which are also cheaper to compute.
The non-extendability criterion of Lemma 8 sharpens the existing sufficient condition and gives a template for studying extension of even-distance binary codes.
Even in the cases where the bound is not tight, the gap of at most 2 in dimension 5 implies the bound is a reliable approximation for practical code-table computations.

Reading between the lines

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

The same recursion should run for $q>2$: the bounds in Theorems 4-8 are written for general prime powers, so constructing the analogous dimension-3/4/5 AFER-optimal databases over non-binary fields is the natural next step.
The uniform gap of at most 2 suggests that an exact formula for $e(n,5,2)$ may exist; testing whether the exceptional codes form finitely many families that stabilize as $s$ grows would give evidence for the paper's first open question.
Since Theorem 9's strength depends on how much is known about lower-dimensional codes, one useful stress test is to verify Proposition 2's subcode-chain existence computationally for dimension 6 before trusting the higher-dimensional bounds.

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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 iterative lower bounds in Theorems 4–8, packaged as Theorem 9, are a genuine step forward for determining AFER-optimal codes. The idea of bootstrapping error-coefficient bounds from lower-dimensional AFER-optimal codes is clean and, based on the tables, gives tight or near-tight results for most binary Griesmer-optimal codes of dimension up to 5. The paper also does something useful beyond the bounds: it completes the catalog of binary AFER-optimal codes in dimension 5, with explicit generator matrices and constructions. Lemma 8, improving Maruta's non-extendability condition for even-distance binary codes, looks like a solid standalone observation.

The soft spots are real but not evenly distributed. The proofs of Theorems 4–8 are terse; the step from (14) to (15) in Theorem 4, in particular, needs a clearer justification of the inequality and the divisibility transformations. I also found Proposition 2's existence argument non-constructive enough that I would want a more careful write-up. These are fixable in revision.

The bigger problem is the pair of exhaustive searches that carry the main completeness claims. In Theorem 11, Case B, the claim that puncturing two [40,5,20] codes yields only [38,5,18;5] is asserted in one sentence. In Theorem 13, the error coefficients 13, 11, and 8 come from "iterating over all feasible solutions" with no code, no case counts, and no description of the search space. These are not auxiliary checks; they are the entire justification for the exact AFER-optimality of four families and for the "gap ≤ 2" statement. As written, a reader cannot verify that the traversal was exhaustive or that the minimum was computed correctly. The stress-test note is right to flag this as the load-bearing weakness.

The paper is not circular and the authors are not hiding the ball—they say explicitly that they used Magma and traversal, and the appendix does provide generator matrices. But for a claim of this strength—"we determine the parameters of all binary 5-dimensional AFER-optimal linear codes"—one-line prose is not enough. The authors should be asked to supply the computational scripts, the exact enumeration used, and ideally a verification script that recomputes the error coefficients of the claimed optimal codes.

Who is this for? Coding theorists working on error coefficients, AFER-optimality, and optimal linear codes; also people maintaining code tables. It deserves a serious referee, but the referee should demand a reproducible computational appendix before the completeness claims are accepted.

Referee Report

4 major / 5 minor

Summary. The paper introduces five iterative lower bounds on the error coefficient (number of minimum-weight codewords) of Griesmer-optimal linear codes, derived from residual codes, subcode chains, and the rank of associated minihypers. The bounds are collected in Theorem 9 and applied to binary codes of dimension up to 5. The authors claim the bounds are tight for most parameters and, for the remaining dimension-5 cases, are within 2 of the true value. They further claim AFER-optimality for several explicit families, most notably [31s2+7,5,16s2+2;5]2 (Theorem 11) and [31s1+8,5,16s1+3;13]2, [31s1+12,5,16s1+5;11]2, [31s2+20,5,16s2+9;8]2 (Theorem 13), and they provide generator matrices for the constructed codes.

Significance. If correct, the paper gives a practically useful method that is simpler and often tighter than the linear programming bound, and it completes the classification of binary AFER-optimal linear codes for k≤5 with explicit constructions. The paper is well organized, contains concrete examples, and supplies generator matrices in an appendix. The main caveats are that the exact AFER-optimality results for the four families in Theorem 11 and Theorem 13 rest on finite exhaustive searches that are not documented reproducibly, and one of the family-level proofs appears to check only a single parameter value. These issues affect the central completeness claim and the claimed gap of at most 2 in Table IV.

major comments (4)

[Theorem 11, Case B (Section IV-A)] The proof of Case B only performs the puncturing traversal on the s2=1 representatives [40,5,20]2 and [38,5,18]2, but Theorem 11 claims AFER-optimality of the entire family [31s2+7,5,16s2+2;5]2 for all s2≥1. No argument shows that the conclusion for s2=1 transfers to s2≥2. Since this family is one of the four claimed AFER-optimal families in Table IV, the proof is incomplete as written.
[Theorem 11, Case B, and Theorem 13 (Section IV)] The assertions 'After traversal puncturing two [40,5,20]2 linear codes... we obtain only [38,5,18;5]2 linear code' and 'After iterating over all feasible solutions, we obtain e(Ĉ′1) ≥ 13, e(Ĉ′2) ≥ 11, and e(Ĉ′3) ≥ 8' are the sole justification for the exact error coefficients 5, 13, 11, and 8. No enumeration algorithm, correctness argument, case table, or code is supplied, so a reader cannot verify that the search was exhaustive or that the minima were computed correctly. Because these values underpin the 'gap ≤ 2' claim in Table IV, the manuscript should provide a reproducible script or a complete, checkable case analysis.
[Theorem 7, Eq. (33) (Section III-D)] The 'constrains relationship' displayed in Eq. (33) is not a valid derivation as typeset: it appears to relate the ratio (Ad(C) - e(gq(k2,d),k2,q)) / e(n - gq(k2,d), k-k2, q) to ςq(k2,d)/(q-1) without proving that the punctured code Υ_A(C) has minimum distance exactly ⌈d/q^{k2}⌉ and is distance-optimal with the required parameters, or that the code attaining e(n-gq(k2,d), k-k2, q) can be realized as a subcode of C. The definitions of the divisibility corrections Δ and Δ′ are also only stated without derivation. The bound may be correct, but the proof needs to be rewritten as an explicit inequality chain.
[Proposition 2 (Section III-C)] In Case A of Proposition 2, the step claiming that an [n-d, k-1, ⌈d/q⌉+1] code 'violates the Griesmer bound' and therefore that some codeword c2 must attain n(⟨c1,c2⟩)=gq(2,d) is too quick. The existence of c2 requires an argument that if all residual codewords had weight at least ⌈d/q⌉+1, then the residual code would be an [n-d, k-1, ≥⌈d/q⌉+1] code whose parameters contradict the Griesmer bound. This can likely be completed, but as written it is a non-constructive existence claim that should be spelled out.

minor comments (5)

[Proposition 2 proof] There is a typo 'theh [n − d, k− 1, ⌈d/q⌉ + 1]q linear code' that should read 'the [n − d, k− 1, ⌈d/q⌉ + 1]q linear code'.
[Table III] In the row for length 31s1+11, the construction is written as '([s2·P[5]], G[42,5,20;3]2)' but the length depends on s1; the subscript should presumably be s1, and similarly in a few other rows the construction mixes s1 and s2.
[Table III] The row for length 31s2+22 uses the symbol 'P′Tk' but the subscript k is undefined; it should be 'P′T4' or a definition should be given.
[Theorem 4, Eq. (14)] The use of Theorem 3 to assert e(gq(2,d),2,q)=(q-1)(t+1) when q|d is not covered by the statement of Theorem 3, since the Griesmer 2-dimensional code with length gq(2,d) is not of the form in Theorem 3 when t=0. The formula is true, but the proof should either cite a direct computation or extend Theorem 3.
[Lemma 8] The proof of Lemma 8 is very compressed; in particular, the equivalence between extendability and the existence of a weight-d codeword c′ with wt(c+c′)=d is asserted without a full argument in both directions. A reference or a few clarifying sentences would help.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the iterative bounds recurse strictly downward in dimension, and tightness is certified by explicit constructions rather than assumed; the remaining 5D cases rest on finite enumerations and a minor self-citation, which are verification concerns, not circular reductions.

full rationale

Walking the derivation chain, the recursion is well-founded rather than circular. Theorem 3 establishes the two-dimensional base case by deriving the unique weight enumerator of the relevant MDS codes and by a recursion on juxtaposed Simplex codes; it does not assume any of the later bounds. Theorems 4 through 8 express e(n,k,q) for dimension k in terms of e or e_d values at dimensions at most k-1: residual codes in Theorem 4, punctured residual codes in Theorem 5, subcodes C_{k2} with k2 < k1 <= k-1 in Theorems 6 and 7, and minihyper ranks k' < k in Theorem 8. Thus every use of a lower-dimensional value terminates at the k=2 base case, so there is no same-level self-dependency. The claimed tightness in Tables I-III is not a fit: for every row the paper supplies an explicit generator matrix or point multi-set, computes its error coefficient directly, and then the corresponding lower bound certifies AFER-optimality. The residual 5-dimensional cases in Section IV depend on Lemma 8, the external classification of weight-2 anti-expansion Griesmer codes [29,36], and two finite enumerations described only in prose (Theorem 11, Case B and Theorem 13). These enumerations are not independently reproducible from the paper, but incompleteness or possible miscomputation would be a verification or correctness gap, not a circular reduction: the AFER-optimality conclusions are not used as inputs to the bounds that produce them. Table IV also cites [26], by two of the present authors, for the two non-Griesmer codes of lengths 11 and 13; this is a minor self-citation used for completeness of the dimension-5 listing, but those values come from a separate published classification and are not used to define or evaluate the iterative bounds. No step was found where a prediction equals an input by construction, or where a fitted parameter is renamed as a prediction, or where a load-bearing argument reduces to an unverified self-citation.

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

The paper does not introduce new physical or mathematical entities. It relies on existing finite geometry constructs and classification results.

assumptions (4)

standard math Griesmer bound and modified Griesmer bound are valid.
Used as the basis for optimality, cited from [2] and [30].
domain assumption Classification results on Griesmer codes with anti-expansion vector weight 2 (SS and Belov type) are correct.
Used in Lemma 4 and Theorems 12 and 13, relies on [29] and [35].
domain assumption The computational classifications of binary linear codes up to length 13 and dimension 5 are complete.
Used to assert all AFER-optimal codes are covered, based on prior work [26], [27].
ad hoc to paper The existence of subcodes with prescribed effective lengths (Proposition 2) holds for all Griesmer optimal codes.
This is a new structural claim whose proof is not fully rigorous; it is used as a foundation for Theorems 6 and 7.

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Pith. "Pith review of Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration." pith.science (2026). https://pith.science/paper/AGVC334H

@misc{pith2026250705567,
  author       = {Pith},
  title        = {Pith review of: Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGVC334H}},
  note         = {Machine review of arXiv:2507.05567}
}

abstract

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed $5$. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to $2$.

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2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Maximum distance q-ary codes,

R. C. Singleton, “Maximum distance q-ary codes,” IEEE Transactions on Information Theory , vol. 10, p. 116–118, 1964

work page 1964

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A bound for error-correcting codes,

J. H. Griesmer, “A bound for error-correcting codes,” IBM Journal of Research and Development , vol. 4, no. 5, pp. 532–542, 1960

work page 1960

[3] [3]

Binary codes with specified minimum distance,

M. Plotkin, “Binary codes with specified minimum distance,” IRE Transactions on Information Theory , vol. 6, no. 4, pp. 445–450, 1960

work page 1960

[4] [4]

Bounds for unrestricted codes, by linear programming,

P. Delsarte, “Bounds for unrestricted codes, by linear programming,” Philips Res. Rep. , vol. 27, p. 272–289, 1972

work page 1972

[5] [5]

Two families of optimal linear codes and their subfield codes,

Z. Heng, Q. Wang, and C. Ding, “Two families of optimal linear codes and their subfield codes,” IEEE Transactions on Information Theory , vol. 66, no. 11, pp. 6872–6883, 2020

work page 2020

[6] [6]

Two conjectures on the largest minimum distances of binary self-orthogonal codes with dimension 5,

M. Shi, S. Li, and J.-L. Kim, “Two conjectures on the largest minimum distances of binary self-orthogonal codes with dimension 5,” IEEE Transactions on Information Theory , vol. 69, no. 7, pp. 4507–4512, 2023

work page 2023

[7] [7]

The hull of two classical propagation rules and their applications,

Y . Li, S. Zhu, and E. Mart ´ınez-Moro, “The hull of two classical propagation rules and their applications,” IEEE Transactions on Information Theory , vol. 69, no. 10, pp. 6500–6511, 2023. 15

work page 2023

[8] [8]

New constructions of optimal linear codes from simplicial complexes,

Z. Hu, Y . Xu, N. Li, X. Zeng, L. Wang, and X. Tang, “New constructions of optimal linear codes from simplicial complexes,” IEEE Transactions on Information Theory, vol. 70, no. 3, pp. 1823–1835, 2024

work page 2024