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arxiv: 1906.12017 · v1 · pith:7LWUORMSnew · submitted 2019-06-28 · 💻 cs.IT · math.CO· math.IT

Binary optimal linear codes from posets of the disjoint union of two chains

Yansheng Wu , Jong Yoon Hyun , Qin Yue This is my paper

Pith reviewed 2026-05-25 14:01 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT
keywords binary linear codesoptimal codesposetsdisjoint union of chainsminimum distancecoding theory
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The pith

Posets from the disjoint union of two chains produce binary optimal linear codes.

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

The paper constructs binary linear codes by using posets that consist of two disjoint chains. These codes are shown to be optimal, satisfying bounds such as the Singleton bound on minimum distance. The method provides an alternative to constructions based on simplicial complexes. Optimal codes maximize error-correcting capability for a given length and dimension. This yields explicit families of codes with verifiable parameters.

Core claim

By defining linear codes from posets that are the disjoint union of two chains, the resulting binary codes achieve optimality in their minimum distance relative to length and dimension.

What carries the argument

The poset of the disjoint union of two chains, which structures the supports or generator matrix to ensure the code meets an optimality bound.

Load-bearing premise

The specific choice of two-chain poset must produce a code whose minimum distance actually reaches the theoretical upper bound for its parameters.

What would settle it

An explicit computation of the minimum Hamming weight for a code from a two-chain poset of given lengths that falls below the Singleton bound value for that dimension and length.

Figures

Figures reproduced from arXiv: 1906.12017 by Jong Yoon Hyun, Qin Yue, Yansheng Wu.

Figure 1
Figure 1. Figure 1: m m − 121 m + 2 m + 1 nn − 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
read the original abstract

Recently, Chang and Hyun obtained some classes of binary optimal codes via simplicial complexes. In this letter, we utilize posets of the disjoint union of two chains to construct binary optimal 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

0 major / 2 minor

Summary. The manuscript constructs binary linear codes from posets that are the disjoint union of two chains. Parameters (length n, dimension k, minimum distance d) are derived directly from the order ideals of these posets; the resulting codes are shown to meet the Griesmer bound with equality for the stated infinite families, establishing optimality.

Significance. The explicit combinatorial construction yields new families of optimal binary codes, extending the simplicial-complex approach of Chang and Hyun. Deriving n, k, d from order ideals and verifying equality in the Griesmer bound supplies concrete, falsifiable examples that can be checked against known tables of optimal codes.

minor comments (2)
  1. The abstract and introduction cite Chang and Hyun but do not list the precise reference; add the full citation in the bibliography.
  2. Notation for the two chains (e.g., lengths m1 and m2) is introduced without an explicit diagram; a small figure of the poset would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper gives an explicit combinatorial construction of binary linear codes whose length, dimension, and minimum distance are read off directly from the order ideals of the poset that is the disjoint union of two chains. These parameters are compared to the external Griesmer bound and shown to meet it with equality for the stated families. No parameter is fitted to data and then re-used as a prediction, no definition is circular, and the single cited prior result (Chang-Hyun) is by different authors and functions only as background motivation. The derivation therefore stands on its own combinatorial definitions and an independent bound.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5546 in / 897 out tokens · 35990 ms · 2026-05-25T14:01:44.641571+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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