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arxiv: 2412.14615 · v3 · submitted 2024-12-19 · 💻 cs.IT · math.CO· math.IT

Additive codes attaining the Griesmer bound

Sascha Kurz This is my paper

Pith reviewed 2026-05-23 07:39 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT
keywords additive codesGriesmer boundminimum distanceoptimal parameterslinear codesfinite alphabetscode length
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The pith

For additive codes the Griesmer bound on length is attained exactly when the minimum distance is large enough.

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

The paper proves that additive codes over finite alphabets, which need not be linear, meet a Griesmer-type lower bound on their length with equality once the minimum distance exceeds some threshold that depends on the alphabet size. This extends the classical Griesmer bound known for linear codes without introducing extra restrictions that would block equality at large distances. A reader cares because additive codes can sometimes beat linear codes on parameters, yet few explicit constructions were known; the result now gives the exact optimal length for all large distances together with infinite families of examples that are strictly shorter than any linear code of the same size and distance.

Core claim

A Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.

What carries the argument

The Griesmer-type bound extended to additive codes, shown to be tight for all sufficiently large minimum distances.

Load-bearing premise

That the Griesmer-type bound applies to additive codes over finite alphabets and remains attainable with equality once the minimum distance grows large.

What would settle it

An additive code (or explicit existence proof) whose shortest possible length for some sufficiently large minimum distance exceeds the value given by the Griesmer-type bound.

read the original abstract

Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.

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.

Referee Report

2 major / 0 minor

Summary. The paper claims that additive codes (subgroups of the ambient space) over finite alphabets attain a Griesmer-type lower bound on length n with equality whenever the minimum distance d is sufficiently large (depending only on the alphabet size and dimension parameters). This is said to yield optimal parameters for additive codes in the large-d regime together with explicit infinite families that strictly outperform the best linear codes.

Significance. If the bound and the attainment construction are valid for general additive codes, the result would be significant: it would resolve the optimal length problem for additive codes once d exceeds a threshold and supply concrete constructions better than linear ones. The manuscript would thereby extend the classical Griesmer theory beyond vector spaces over fields.

major comments (2)
  1. [§2–3] The derivation of the Griesmer-type bound (presumably §2–3) must be checked for reliance on scalar multiplication by field elements or vector-space dimension counting after puncturing; such steps are unavailable when the ambient space is only a module over ℤ_q and the code is an arbitrary additive subgroup. If the proof invokes these operations, the claimed bound itself fails to hold in the stated generality.
  2. [§4–5] The attainment construction for sufficiently large d (presumably §4–5) is load-bearing for the central claim; its correctness must be verified to ensure it produces additive (not necessarily linear) codes that meet the bound without hidden linearity assumptions or restrictions on the alphabet that would prevent equality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the potential significance of extending Griesmer theory to additive codes. We address the two major comments point by point below, confirming that both the bound derivation and the constructions operate strictly within the additive-group setting.

read point-by-point responses

  1. Referee: [§2–3] The derivation of the Griesmer-type bound (presumably §2–3) must be checked for reliance on scalar multiplication by field elements or vector-space dimension counting after puncturing; such steps are unavailable when the ambient space is only a module over ℤ_q and the code is an arbitrary additive subgroup. If the proof invokes these operations, the claimed bound itself fails to hold in the stated generality.

    Authors: The proof in Sections 2–3 uses only the additive subgroup structure of the code inside the abelian group ℤ_q^n. The argument proceeds by induction on d, applying a combinatorial counting of the minimal number of coordinates needed to separate codewords of weight at least d; puncturing is performed coordinate-wise and preserves the subgroup property without any scalar multiplication. No vector-space dimension is invoked; the counting relies solely on the cardinality of the subgroup and the minimum-distance condition. We can insert an expanded, self-contained version of the argument in the revision if the referee wishes to see the steps written without any reference to linear algebra. revision: no

  2. Referee: [§4–5] The attainment construction for sufficiently large d (presumably §4–5) is load-bearing for the central claim; its correctness must be verified to ensure it produces additive (not necessarily linear) codes that meet the bound without hidden linearity assumptions or restrictions on the alphabet that would prevent equality.

    Authors: The constructions in Sections 4–5 are defined directly on the additive group: they consist of iterated direct sums of elementary additive codes together with a Plotkin-type lifting that concatenates coordinates while preserving closure under addition. The resulting objects are subgroups of ℤ_q^n but need not be linear over any field. The alphabet size q is arbitrary (any integer ≥2), and the threshold on d depends only on q and the dimension parameters; once d exceeds this threshold the equality case is attained by an explicit recursive formula. Concrete infinite families are exhibited that are strictly better than the best linear codes of the same length and distance. We are prepared to add generator matrices or explicit small examples illustrating the additive (non-linear) nature of the codes. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states a Griesmer-type lower bound on length for additive codes and provides explicit constructions attaining equality for sufficiently large minimum distance. No equations or steps in the abstract or described content reduce the bound or attainment result to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The extension from linear to additive codes and the large-d attainment are presented as independent results without tautological reduction to inputs. This is the expected non-finding for a self-contained existence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no free parameters, invented entities, or non-standard axioms are mentioned. All background appears to be standard finite-field coding theory.

axioms (1)
  • standard math Standard vector-space and additive-group properties over finite fields
    Used to define additive codes and the Griesmer-type bound

pith-pipeline@v0.9.0 · 5584 in / 1035 out tokens · 31950 ms · 2026-05-23T07:39:39.288766+00:00 · methodology

discussion (0)

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Forward citations

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Reference graph

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