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arxiv: 1907.09644 · v1 · pith:3PRL57TDnew · submitted 2019-07-23 · 🧮 math.CO · math.AC

Hamming Polynomial of a Demimatroid

Jose Martinez-Bernal , Miguel A. Valencia-Bucio , Rafael H. Villarreal This is my paper

Pith reviewed 2026-05-24 17:59 UTC · model grok-4.3

classification 🧮 math.CO math.AC
keywords demimatroidHamming polynomialTutte polynomialBetti numberscombinatroidsimplicial complexWei dualitycoding theory
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The pith

The Hamming polynomial of a demimatroid is equivalent to its Tutte polynomial.

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

The paper introduces the Hamming polynomial for demimatroids as an extension of the Hamming weight enumerator from matroids. It establishes that this polynomial is both a specialization of and equivalent to the Tutte polynomial of the demimatroid. The authors further prove that the Betti numbers associated with a demimatroid and its elongations fully determine the Hamming polynomial. This framework extends to simplicial complexes and even to the more general combinatroids, allowing coding theory concepts like the MacWilliams identity to transfer directly. The work provides evidence that demimatroids are suitable objects for generalizing duality results in coding theory.

Core claim

We define the Hamming polynomial W(x,y,t) of a demimatroid M as a generalization of the extended Hamming weight enumerator of a matroid. The polynomial W(x,y,t) is a specialization of the Tutte polynomial of M, and actually is equivalent to it. Guided by work on matroids, we prove that Betti numbers of a demimatroid and its elongations determine the Hamming polynomial. These results apply to simplicial complexes and extend to combinatroids where many coding theory invariants carry over.

What carries the argument

The Hamming polynomial W(x,y,t), which generalizes the extended Hamming weight enumerator and is shown to be equivalent to the Tutte polynomial for demimatroids.

If this is right

  • The Hamming polynomial can be determined using Betti numbers of the demimatroid and its elongations.
  • Results apply directly to simplicial complexes viewed as demimatroids.
  • Concepts such as the Tutte polynomial, characteristic polynomial, MacWilliams identity, and generalized Hamming polynomials extend to combinatroids.
  • Deletion, contraction, and duality extend to these non-matroidal structures.

Where Pith is reading between the lines

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

  • The equivalence suggests that properties of the Tutte polynomial can be used to study Hamming weights in demimatroids without separate computation.
  • The extension to combinatroids indicates that coding invariants may apply to any integer-valued function on subsets with value zero on the empty set.
  • The conjectural part on the r-th generalized Hamming polynomial could be checked on small explicit combinatroids to test the pattern.

Load-bearing premise

That demimatroids as defined following Britz et al. are the appropriate objects for studying Wei's duality and that the stated equivalences hold under the generalizations to combinatroids.

What would settle it

A counterexample demimatroid in which the Hamming polynomial W(x,y,t) differs from the Tutte polynomial or cannot be recovered from the Betti numbers of the demimatroid and its elongations.

read the original abstract

Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi\-ma\-troids as a(nother) natural generalization of matroids. As they have shown, demi\-ma\-troids are the appropriate combinatorial objects for studying Wei's duality. Our results here apport further evidence about the trueness of that observation. We define the Hamming polynomial of a demimatroid $M$, denoted by $W(x,y,t)$, as a generalization of the extended Hamming weight enumerator of a matroid. The polynomial $W(x,y,t)$ is a specialization of the Tutte polynomial of $M$, and actually is equivalent to it. Guided by work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of a demimatroid and its elongations determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to combinatroids. A combinatroid, or Brylawski structure, is an integer valued function $\rho$, defined over the power set of a finite ground set, satisfying the only condition $\rho(\emptyset)=0$. Even in this extreme generality, we will show that many concepts and invariants in coding theory can be carried on directly to combinatroids, say, Tutte polynomial, characteristic polynomial, MacWilliams identity, extended Hamming polynomial, and the $r$-th generalized Hamming polynomial; this last one, at least conjecturelly, guided by the work of Jurrius and Pellikaan for linear codes. All this largely extends the notions of deletion, contraction, duality and codes to non-matroidal structures.

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 / 4 minor

Summary. The paper defines the Hamming polynomial W(x,y,t) of a demimatroid M as a generalization of the extended Hamming weight enumerator of a matroid. It claims that W(x,y,t) is a specialization of the Tutte polynomial of M and is in fact equivalent to it. Guided by prior work on matroids, the authors prove that the Betti numbers of a demimatroid together with those of its elongations determine W(x,y,t). The results are applied to simplicial complexes (viewed canonically as demimatroids) and the framework is extended to combinatroids (Brylawski structures satisfying only ρ(∅)=0), where the Tutte polynomial, characteristic polynomial, MacWilliams identity, extended Hamming polynomial, and r-th generalized Hamming polynomial are shown to carry over directly.

Significance. If the central equivalences and determinations hold, the work supplies concrete additional evidence that demimatroids are natural objects for Wei duality and demonstrates that several fundamental invariants from matroid theory and coding theory remain well-defined and interrelated under substantial relaxations of the matroid axioms. The explicit determination of W by Betti numbers (extending Johnsen-Roksvold-Verdure) and the direct carry-over to combinatroids are the most noteworthy contributions.

minor comments (4)
  1. [Abstract] Abstract: 'apport further evidence' is a typographical error; the intended word is 'support'.
  2. [Abstract] Abstract: 'the trueness of that observation' should read 'the truth of that observation'.
  3. [Abstract] Abstract: 'conjecturelly' is a typographical error; the correct spelling is 'conjecturally'.
  4. [Abstract] Abstract: the phrasing 'say, Tutte polynomial, characteristic polynomial...' is informal for a journal submission; a more precise enumeration 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 of minor revision. No specific major comments appear in the report, so there are no individual points requiring detailed rebuttal or changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external priors and explicit proofs

full rationale

The paper defines the Hamming polynomial W(x,y,t) explicitly as a generalization of the matroid case and then proves (not assumes) its equivalence to the Tutte polynomial via specialization. It further proves that Betti numbers of the demimatroid and elongations determine W, citing external guidance from Johnsen-Roksvold-Verdure. Extensions to combinatroids are presented as direct carry-over of invariants (Tutte, MacWilliams, etc.) under the minimal axiom ρ(∅)=0, again without reducing any claimed result to a self-referential fit or prior self-citation. No step equates a derived quantity to its own input by construction, and all load-bearing references are to non-overlapping authors. The central claims therefore remain independent of the paper's own definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on the prior definition of demimatroids from Britz et al. for the duality property and introduces combinatroids by definition only.

axioms (1)
  • domain assumption Demimatroids are the appropriate combinatorial objects for studying Wei's duality (following Britz, Johnsen, Mayhew and Shiromoto).
    Stated in the opening sentence of the abstract as the foundation for the work.
invented entities (1)
  • combinatroid (Brylawski structure) no independent evidence
    purpose: Further generalization of demimatroids to any integer-valued function on the power set with rho(empty set) = 0.
    Defined in the abstract as the extreme generality in which coding invariants are still carried over.

pith-pipeline@v0.9.0 · 5870 in / 1370 out tokens · 23161 ms · 2026-05-24T17:59:55.111460+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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