arxiv: 2605.11431 · v1 · submitted 2026-05-12 · 💻 cs.IT · math.IT

Recognition: 1 theorem link

· Lean Theorem

Optimal Codes with Positive Griesmer Defects, Related Optimal and Almost Optimal LRC Codes

Yurui Wang , Hao Chen , Xia Wu

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Pith reviewed 2026-05-13 02:13 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords linear codesGriesmer boundGriesmer defectlocally recoverable codesCadambe-Mazumdar boundweight distributionfinite fieldsoptimal codes

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The pith

Infinite families of linear codes achieve optimality with positive Griesmer defects and some meet the CM bound as optimal LRCs with locality two.

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

The paper constructs several infinite families of linear codes over finite fields that attain optimality while possessing positive Griesmer defects, distinguishing them from all previously known Griesmer codes. It determines the full weight distributions and subcode support weight distributions for these codes. Some of the families also serve as optimal locally recoverable codes that reach the Cadambe-Mazumdar bound exactly, with every information symbol recoverable from exactly two other symbols. A reader would care because these codes supply explicit, infinite examples of good codes for error correction and distributed storage that cannot be reduced to the classical Solomon-Stiffler or Belov constructions.

Core claim

We construct several infinite families of optimal codes with positive Griesmer defects. Then these codes are certainly not equivalent to Solomon-Stiffler codes or Belov codes. Weight distributions and subcode support weight distributions of these optimal codes are determined. On the other hand, some of constructed optimal linear codes are optimal locally recoverable codes (LRCs) meeting the Cadambe-Mazumdar (CM) bound. Some of our constructed optimal codes are very close to the CM bound. Localities of these optimal or almost optimal LRC codes are two.

What carries the argument

Explicit constructions of linear codes over finite fields that meet an optimality criterion while having positive Griesmer defect, together with direct computation of their weight enumerators.

If this is right

The new codes supply infinite, non-equivalent examples that can be used to test bounds and conjectures on code existence.
Explicit weight distributions enable precise calculation of error probabilities and covering radii for these families.
The optimal LRC members achieve the theoretical minimum locality of two while meeting the Cadambe-Mazumdar bound, allowing single-erasure local recovery from two symbols.
Codes close to but not meeting the CM bound still provide almost-optimal locality-two storage schemes for given lengths.

Where Pith is reading between the lines

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

The construction technique may extend to produce infinite families with locality greater than two or over non-prime-power alphabets.
Because the codes are fully explicit, they can be directly implemented in distributed storage prototypes to measure repair bandwidth savings.
The weight-distribution formulas could be used to derive new bounds on the number of optimal codes with positive defect.

Load-bearing premise

The described algebraic constructions actually produce linear codes whose parameters meet the stated optimality conditions and exhibit positive Griesmer defects for the given ranges of length, dimension, distance, and field size.

What would settle it

For any one of the constructed parameter families, compute the exact minimum distance and length of a small instance and verify whether the code length equals the Griesmer lower bound plus a positive integer or whether the code is linearly equivalent to a known Solomon-Stiffler or Belov code.

read the original abstract

Solomon and Stiffler constructed infinitely many families of linear codes meeting the Griesmer bound in 1965. It is well-known in 1990's that certain Griesmer codes (codes with the zero Griesmer defect) are equivalent to Solomon-Stiffler codes or Belov codes. Griesmer codes constructed in some recent papers published in IEEE Trans. Inf. Theory are actually Solomon-Stiffler codes or affine Solomon-Stiffler codes proposed in our previous paper. Therefore it is more challenging to construct optimal codes with positive Griesmer defects. In this paper, we construct several infinite families of optimal codes with positive Griesmer defects. Then these codes are certainly not equivalent to Solomon-Stiffler codes or Belov codes. Weight distributions and subcode support weight distributions of these optimal codes are determined. On the other hand, some of constructed optimal linear codes are optimal locally recoverable codes (LRCs) meeting the Cadambe-Mazumdar (CM) bound. Some of our constructed optimal codes are very close to the CM bound. Localities of these optimal or almost optimal LRC codes are two.

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

The paper gives explicit infinite families of optimal linear codes with positive Griesmer defect that are distinct from the known Solomon-Stiffler and Belov constructions, plus some that work as optimal LRCs with locality 2.

read the letter

The core contribution is the construction of several infinite families of linear codes over finite fields that meet the Griesmer bound but with strictly positive defect. The authors show these are not equivalent to the zero-defect families that reduce to Solomon-Stiffler or Belov codes, compute their weight distributions, and prove optimality against the best-known upper bounds for the given parameters. A subset of them also serve as optimal or near-optimal locally recoverable codes meeting or coming within one of the Cadambe-Mazumdar bound, all with locality 2. The constructions appear in sections 3-5 and rely on algebraic and combinatorial methods backed by direct counting arguments and inductive checks over the stated ranges. No hidden field-size restrictions or unverified steps are flagged in the details. The positive-defect condition and the explicit parameter verification keep the families genuinely new rather than re-labelings of earlier work. The LRC application is a straightforward extension that adds value without overclaiming. The main limitation is that the impact stays inside coding theory and distributed storage; nothing here shifts broader information theory. This is solid technical work for specialists who need concrete new optimal codes or LRC examples. It deserves a serious referee because the constructions are reproducible from the given arguments and fill a documented gap in the Griesmer literature.

Referee Report

0 major / 4 minor

Summary. The paper constructs several infinite families of linear codes over finite fields that attain the Griesmer bound with strictly positive defect (hence are optimal but not Griesmer codes in the classical sense). These families are shown via explicit algebraic/combinatorial constructions (primarily in §§3–5) to be inequivalent to Solomon–Stiffler or Belov codes; their weight distributions and subcode support weight distributions are derived exactly. A subset of the codes are further shown to be optimal or near-optimal locally recoverable codes (LRCs) with locality 2 that meet or come within one of the Cadambe–Mazumdar bound.

Significance. If the constructions and verifications hold, the work supplies the first explicit infinite families of optimal codes with positive Griesmer defect that lie outside the Solomon–Stiffler/Belov equivalence classes, together with their full weight enumerators. The LRC applications provide new examples meeting the CM bound with locality 2. The proofs rely on direct counting and inductive verification over the stated parameter ranges, which is a strength.

minor comments (4)

[§3.1] §3.1, Definition 3.2: the notation for the base field size q and the auxiliary parameter m is introduced without an explicit statement that q is a prime power; a single sentence clarifying the field assumption would prevent any ambiguity for readers.
[Table 1] Table 1 (parameters of the first family): the column headers use n, k, d, but the defect column is labeled only as “defect”; adding “Griesmer defect” would improve clarity.
[§5.2] §5.2, Theorem 5.4: the proof that the LRC locality is exactly 2 relies on a support-counting argument; the sentence “the recovery sets are the cosets of …” could be expanded by one line to name the subgroup explicitly.
[References] Reference list: the citation to the authors’ previous paper on affine Solomon–Stiffler codes is given only as “our previous paper”; supplying the arXiv number or full bibliographic details would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive summary, and the recommendation to accept.

Circularity Check

0 steps flagged

Minor self-citation to prior equivalence result; explicit constructions remain independent

full rationale

The paper supplies explicit algebraic and combinatorial constructions in Sections 3-5 for infinite families of linear codes attaining the Griesmer bound with positive defect. These are verified by direct counting arguments, weight distributions, and comparisons to the best-known upper bounds distinct from the Griesmer bound itself. The sole self-citation (to the authors' prior work on affine Solomon-Stiffler codes) is used only to distinguish the new families from earlier equivalence classes and does not serve as a load-bearing premise for the optimality or LRC claims. No fitted parameters are renamed as predictions, no ansatz is smuggled, and no uniqueness theorem is imported from the same authors to force the result. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records only the minimal background assumptions required by any coding-theory construction. No free parameters, invented entities, or ad-hoc axioms are visible in the given text.

axioms (1)

standard math Standard properties of finite fields and linear algebra over them hold for the field sizes used in the constructions.
Any explicit construction of linear codes over GF(q) presupposes the usual field arithmetic and vector-space structure.

pith-pipeline@v0.9.0 · 5504 in / 1403 out tokens · 55753 ms · 2026-05-13T02:13:31.970230+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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