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arxiv: 2606.29022 · v1 · pith:UPQOT46J · submitted 2026-06-27 · math.CO

An Explication of Optimal Equidistant Codes

Douglas R. Stinson This is my paper

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classification math.CO
keywords equidistant codesbinary codessymmetric BIBDoptimal codescode characterizationsconstant weight codes
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The pith

Characterizations of optimal equidistant binary codes have overlooked one subcase when n ≡ 2 mod 4.

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

This expository paper examines the long-standing problem of characterizing binary codes of length n that achieve both the largest possible minimum distance and the maximum number of codewords. It collects and unifies results from multiple authors showing that such optimal equidistant codes correspond to symmetric balanced incomplete block designs with specific parameters. The review notes that an early 1973 paper by Marrero and Butson on the topic went unnoticed in subsequent literature. It further points out that existing lists of these codes and designs are incomplete for the congruence class n ≡ 2 mod 4.

Core claim

The paper establishes that the known characterizations of equidistant binary codes of length n with maximum distance and maximum size, which rest on correspondences with symmetric BIBDs, are missing one of the admissible subcases precisely when n ≡ 2 mod 4. It also records that the 1973 work of Marrero and Butson already contained relevant results that later authors did not cite.

What carries the argument

Symmetric balanced incomplete block designs with parameters tied to code length n and distance d, which supply the combinatorial objects whose incidence matrices yield the equidistant codes.

If this is right

  • The maximum number of codewords in an optimal equidistant binary code of length n is determined by the parameters of the corresponding symmetric BIBD.
  • For n ≡ 2 mod 4 an additional family of symmetric BIBDs (or codes) must be considered that was omitted from earlier enumerations.
  • Any complete classification of such codes must incorporate the 1973 Marrero-Butson results alongside later contributions.
  • The distance and size bounds derived from the BIBD parameters remain valid across all congruence classes of n.

Where Pith is reading between the lines

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

  • The missed subcase may correspond to known but unlinked designs or to new parameter sets that admit explicit constructions.
  • Future exhaustive searches for small n ≡ 2 mod 4 could reveal whether the additional case yields codes that improve upon or match existing tables.
  • The oversight suggests that a systematic cross-check between coding-theory and design-theory databases would prevent similar gaps in related problems.

Load-bearing premise

Characterizations of these optimal equidistant codes involve symmetric BIBDs with certain parameters, as studied by several authors over the years.

What would settle it

Explicit construction or non-existence proof of an equidistant binary code achieving the claimed maximum size for a concrete n ≡ 2 mod 4 that falls outside all previously listed BIBD parameter sets would decide whether the missed subcase is real.

Figures

Figures reproduced from arXiv: 2606.29022 by Douglas R. Stinson.

Figure 1
Figure 1. Figure 1: Optimal Equidistant Binary Codes Step 2 in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

We discuss the problem of characterizing equidistant binary codes of a given length $n$ having largest possible distance and the maximum number of codewords. Such characterizations have been studied by several authors over the years and they involve symmetric BIBDs with certain parameters. In this primarily expository paper, we investigate the history of this problem and give a unified presentation of the main results. Perhaps surprisingly, researchers on this problem were unaware of early relevant work by Marrero and Butson from 1973. Also, it turns out that published results on characterizations of equidistant binary codes have missed one of the possible subcases when $n \equiv 2 \bmod 4$.

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

0 major / 2 minor

Summary. The manuscript is an expository review of characterizations of optimal equidistant binary codes of length n (maximum distance and maximum number of codewords). It unifies prior results that connect these codes to symmetric BIBDs with standard parameters, supplies a historical note on the overlooked Marrero-Butson (1973) work, and asserts that published characterizations have missed one subcase when n ≡ 2 mod 4.

Significance. If the historical correction and the identification of the missed subcase are accurate, the paper provides a useful archival reference that prevents repetition of prior oversights in combinatorial coding theory. The unified presentation of BIBD-based characterizations is a modest but positive contribution for researchers consulting the literature; the work contains no new theorems, derivations, or code constructions.

minor comments (2)
  1. The abstract states that one subcase for n ≡ 2 mod 4 has been missed but does not name the subcase or the specific prior works that omitted it; adding a one-sentence clarification would make the central historical claim immediately verifiable without requiring the reader to consult the full BIBD literature.
  2. A compact table listing the parameter regimes (n mod 4) together with the corresponding symmetric BIBD parameters and the associated code parameters would strengthen the claimed unified presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive recommendation to accept the manuscript. The report accurately captures the paper's expository nature, its unification of prior results on equidistant binary codes via symmetric BIBDs, the historical note on Marrero-Butson (1973), and the identification of a missed subcase for n ≡ 2 mod 4.

Circularity Check

0 steps flagged

No significant circularity; purely expository literature review

full rationale

The paper is explicitly expository: it unifies existing characterizations of optimal equidistant binary codes (via symmetric BIBDs) and supplies historical notes plus one overlooked subcase for n ≡ 2 mod 4. No new theorems, no derivations, no parameter fitting, no predictions, and no load-bearing self-citations or ansatzes. The argument rests only on accurate citation of prior established BIBD results. No derivation chain exists to inspect for circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper introduces no new free parameters, axioms, or invented entities; it reviews existing links between equidistant codes and symmetric BIBDs.

pith-pipeline@v0.9.1-grok · 5627 in / 974 out tokens · 39838 ms · 2026-06-30T09:15:44.386581+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

26 extracted references · 2 canonical work pages · 1 internal anchor

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