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arxiv: 1906.09645 · v1 · pith:4ZGA24VVnew · submitted 2019-06-23 · 💻 cs.IT · math.IT

On q-ary codes with two distances d and d+1

P. Boyvalenkov , K. Delchev , D. Zinoviev , V. Zinoviev This is my paper

Pith reviewed 2026-05-25 17:30 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords q-ary codestwo-distance codesequidistant codeslinear codesupper boundsblock codescoding theory
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The pith

q-ary codes with distances d and d+1 arise from modifying equidistant codes

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

The paper examines q-ary block codes whose pairwise distances take only the values d and d+1. It supplies several constructions for these codes. In the linear case it establishes that every such code comes from a simple modification of a linear equidistant code. Upper bounds on the largest possible cardinality are derived, and tables of lower and upper bounds appear for small q and n. A reader would care because the results classify a restricted family of codes and give concrete size limits for given length and alphabet size.

Core claim

The paper claims that q-ary codes with two distances d and d+1 admit several constructions, and that in the linear case all such codes can be obtained by a simple modification of linear equidistant codes, while upper bounds are derived for their maximum cardinality.

What carries the argument

Simple modification of linear equidistant codes, which produces all linear examples with distances d and d+1

If this is right

  • All linear codes with these distances are generated by the modification.
  • Upper bounds apply directly to limit maximum cardinality.
  • Constructions work for both linear and nonlinear codes.
  • Tables give explicit bounds for small q and n.

Where Pith is reading between the lines

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

  • The modification may preserve algebraic structure that aids decoding algorithms.
  • Similar modifications could classify codes with three or more distances.
  • The bounds may be achieved by known combinatorial objects such as designs.

Load-bearing premise

That every linear code with distances d and d+1 arises from the described simple modification of an equidistant linear code.

What would settle it

A linear q-ary code with distances only d and d+1 that cannot be obtained from any equidistant linear code by the modification.

read the original abstract

The $q$-ary block codes with two distances $d$ and $d+1$ are considered. Several constructions of such codes are given, as in the linear case all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for the maximum cardinality of such codes is derived. Tables of lower and upper bounds for small $q$ and $n$ are presented.

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 / 1 minor

Summary. The paper studies q-ary block codes having exactly two distances d and d+1. It supplies several constructions, states that every linear example arises from a simple modification of a linear equidistant code, derives upper bounds on maximum cardinality, and tabulates lower/upper bounds for small q and n.

Significance. A rigorous classification of linear two-distance codes together with matching bounds would be a modest but useful addition to the literature on constant-distance and two-weight codes; the tables could serve as a reference for small-parameter searches. The significance is limited by the absence of explicit definitions and derivations for the central linear-case claim.

major comments (2)
  1. [Abstract] Abstract (sentence 2): the assertion that 'in the linear case all codes can be obtained by a simple modification of linear equidistant codes' is load-bearing for the classification result yet supplies neither a definition of the modification operation nor a proof that every linear (d,d+1)-code is reached; the claim therefore cannot be verified from the manuscript.
  2. [Abstract] Abstract: the upper-bound derivations are announced but no explicit bound expressions, proof sketches, or comparison with known Singleton/Plotkin-type bounds appear; without these steps the claimed improvement cannot be assessed.
minor comments (1)
  1. [Abstract] Abstract: 'Upper bounds for the maximum cardinality of such codes is derived' contains a subject-verb agreement error ('bounds … is').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract] Abstract (sentence 2): the assertion that 'in the linear case all codes can be obtained by a simple modification of linear equidistant codes' is load-bearing for the classification result yet supplies neither a definition of the modification operation nor a proof that every linear (d,d+1)-code is reached; the claim therefore cannot be verified from the manuscript.

    Authors: We agree the abstract does not define the modification or include the proof. The body of the manuscript defines the modification (a coordinate-wise shift by a fixed vector applied to an equidistant code) and states the classification, but the referee is correct that an explicit proof is not provided. We will add both a precise definition and a short proof in a new subsection of the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract: the upper-bound derivations are announced but no explicit bound expressions, proof sketches, or comparison with known Singleton/Plotkin-type bounds appear; without these steps the claimed improvement cannot be assessed.

    Authors: The manuscript derives the bounds in Section 4 but presents them only as statements without formulas or comparisons in the abstract. We will insert the explicit bound expressions (a two-distance Plotkin-type bound and a linear-programming bound), a one-paragraph proof sketch, and a direct comparison with the Singleton and Plotkin bounds into both the abstract and the main text of the revision. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions and bounds presented as independent results

full rationale

The abstract states that linear two-distance codes arise via simple modification of equidistant linear codes and that upper bounds are derived, but provides no equations, parameter fits, or self-citations that reduce any claimed result to its own inputs by construction. The modification is described as a construction rather than a definitional equivalence, and no load-bearing self-citation or ansatz smuggling is present. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5595 in / 1059 out tokens · 23373 ms · 2026-05-25T17:30:12.407688+00:00 · methodology

discussion (0)

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Reference graph

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