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arxiv: 2606.17300 · v1 · pith:2FTKXDZRnew · submitted 2026-06-15 · 💻 cs.IT · math.IT

Construction of codes over a commutative non-unital ring from simplicial complexes and their applications

Vidya Sagar , Shikha Patel , Sanjay Kumar Singh This is my paper

Pith reviewed 2026-06-27 02:19 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords linear codes over ringssimplicial complexesGray imagesdivisible codesminimal codeslocally recoverable codesstrongly regular graphsself-orthogonal codes
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The pith

Linear codes over a non-unital ring S built from simplicial complexes with multiple maximal elements have Gray images and subfield-like versions that form divisible, minimal, and self-orthogonal families.

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

The paper constructs linear codes over the ring S, an extension of a commutative non-unital ring I of order p squared, by taking defining sets from general simplicial complexes that may contain several maximal elements. Parameters of the codes are determined explicitly, and both their Gray images and the corresponding subfield-like codes are examined in detail. These images and subcodes are shown to produce multiple families of divisible codes. Under stated sufficient conditions the codes become minimal, optimal, and self-orthogonal. The resulting objects are then applied to obtain projective few-weight codes, locally recoverable codes with small locality, minimal access structures for secret-sharing schemes, and strongly regular graphs with explicit parameters.

Core claim

Defining sets taken from general simplicial complexes that may contain multiple maximal elements determine linear codes over the ring S; the Gray images of these codes and their subfield-like versions are divisible, and under sufficient conditions they are minimal, optimal, and self-orthogonal, which directly yields families of projective few-weight codes, locally recoverable codes with small locality, and strongly regular graphs from the projective two-weight cases.

What carries the argument

Defining sets extracted from general simplicial complexes (possibly with multiple maximal elements) that generate linear codes over the ring S, followed by the Gray map and subfield-like code constructions.

If this is right

  • Gray images and subfield-like codes produce several families of divisible codes.
  • Under sufficient conditions the codes are minimal, optimal, and self-orthogonal.
  • Projective few-weight codes and locally recoverable codes with small locality are obtained.
  • Duals of the minimal codes determine minimal access structures for secret-sharing schemes.
  • Projective two-weight codes produce strongly regular graphs whose parameters are determined explicitly.

Where Pith is reading between the lines

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

  • The allowance for multiple maximal elements in the simplicial complexes enlarges the set of available defining sets beyond those limited to a single maximal element.
  • Self-orthogonality of the constructed codes supplies a direct route to new examples that satisfy the conditions needed for quantum stabilizer codes.
  • The explicit parameters of the resulting strongly regular graphs can be matched against existing tables to identify new isomorphism classes.

Load-bearing premise

Defining sets from simplicial complexes that contain multiple maximal elements produce linear codes over S whose parameters, divisibility, minimality, and optimality match the stated claims.

What would settle it

Pick one concrete simplicial complex with two distinct maximal elements, compute the resulting linear code over S and its Gray image, then check whether the minimum distance and weight distribution satisfy the claimed divisibility and minimality conditions.

read the original abstract

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

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 / 3 minor

Summary. The manuscript constructs linear codes over the commutative non-unital ring S (an extension of a ring I of order p²) via the defining-set method, where the defining sets come from general simplicial complexes that may contain multiple maximal elements. It determines the parameters of the resulting codes over S, studies their Gray images and the corresponding subfield-like codes, proves that these produce families of divisible codes, and gives sufficient conditions under which the codes are minimal, optimal, and self-orthogonal. Applications include families of projective few-weight codes, locally recoverable codes with small locality, minimal access structures for secret-sharing schemes from the duals of minimal codes, and strongly regular graphs obtained from projective two-weight codes, with explicit parameter determinations.

Significance. If the parameter calculations and minimality/optimality conditions hold, the work supplies new explicit families of codes over a non-unital ring, a setting that receives less attention than unital rings, together with direct links to combinatorial designs (strongly regular graphs) and applications (LRCs and secret-sharing). The use of simplicial complexes with multiple maximal faces as defining sets is a natural extension of prior work and could be of interest to researchers working at the intersection of coding theory and combinatorial topology.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3: the claimed weight distribution of the Gray image code appears to rely on the assumption that the simplicial complex has no repeated maximal faces in the support of the defining set; when multiple maximal elements are allowed (as stated in the abstract), the multiplicity counting in the weight formula needs an explicit correction term that is not visible in the displayed expression.
  2. [§5.2, Proposition 5.7] §5.2, Proposition 5.7: the self-orthogonality condition is stated in terms of the inner product over S, but the proof sketch only verifies it for the subfield-like subcode; the extension to the full Gray image requires an additional verification that the Gray map preserves the required orthogonality relations, which is not supplied.
minor comments (3)
  1. [§2] The notation for the ring extension S over I is introduced without an explicit presentation or multiplication table; a short table or basis description in §2 would improve readability.
  2. [Tables 1–3] Several parameter tables list only the length and dimension; adding the minimum distance column (even when it follows from the weight enumerator) would make the optimality claims easier to check at a glance.
  3. [§6] The definition of 'projective' code in the application section should be restated explicitly, as the term is used both for the code and for the associated projective geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific comments on Theorems 4.3 and Proposition 5.7. We address each point below.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3: the claimed weight distribution of the Gray image code appears to rely on the assumption that the simplicial complex has no repeated maximal faces in the support of the defining set; when multiple maximal elements are allowed (as stated in the abstract), the multiplicity counting in the weight formula needs an explicit correction term that is not visible in the displayed expression.

    Authors: The simplicial complexes used in the construction are standard set-theoretic objects whose faces (including maximal faces) are distinct by definition; the phrase “multiple maximal elements” refers to the presence of several distinct maximal faces rather than repeated instances of the same face. Consequently the support of each defining set is a set without multiplicity greater than one, and the weight formula in Theorem 4.3 already sums over these distinct faces. We will add a short clarifying sentence in §4 stating that the support is a simple set and briefly recalling the counting argument, thereby making the absence of an extra correction term explicit. revision: partial

  2. Referee: [§5.2, Proposition 5.7] §5.2, Proposition 5.7: the self-orthogonality condition is stated in terms of the inner product over S, but the proof sketch only verifies it for the subfield-like subcode; the extension to the full Gray image requires an additional verification that the Gray map preserves the required orthogonality relations, which is not supplied.

    Authors: The referee is correct that the current proof sketch verifies self-orthogonality only for the subfield-like subcode. We will expand the proof of Proposition 5.7 to include the missing step: we show that the Gray map is a linear isometry that sends pairs of orthogonal vectors over S to orthogonal vectors over the image alphabet, thereby extending the self-orthogonality statement to the full Gray image. The revised proof will appear in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper describes a standard defining-set construction of linear codes over the ring S derived from simplicial complexes (possibly with multiple maximal elements), followed by analysis of Gray images and subfield-like subcodes. Parameters, divisibility, minimality, optimality, and self-orthogonality are derived directly from the algebraic properties of the chosen defining sets and the ring structure. No equations, parameter-fitting procedures, or self-citations are visible that reduce any claimed result to an input by construction. The listed applications (projective few-weight codes, LRCs, secret-sharing schemes, strongly regular graphs) follow as consequences of the base code properties without circular reduction. The derivation chain is self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; all technical details rest on unstated prior results in coding theory.

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Reference graph

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