arXiv subjects

Koji Imamura

Publications and source records attributed to Koji Imamura.

4 recordsLinked to original sources

arXiv2026

Periodicity of weight enumerators for codes generated by an integral matrix

In the theory of error-correcting codes, the minimum weight and the weight enumerator play a crucial role in evaluating the error-correcting performance. In this paper, by viewing the weight enumerator as a quasi-polynomial, we reduce the determination of the minimum weight of the resulting family to finitely many representative values of $q$. We also give a transformation formula between the Tutte quasi-polynomial and the weight enumerator. Furthermore, we compute the number of full-support codewords for the codes related to the special matroids $N_k$ and $Z_k$. This is equivalent to computing the characteristic quasi-polynomials of the hyperplane arrangements related to $N_k$ and $Z_k$.

math.CO↗

arXiv2026

Higher Rank-Support Weights and q-Polymatroids

The aim of this paper is to develop a $(q,m)$-polymatroidal approach to higher supports and higher rank-weight enumerators of rank-metric codes. In this framework, we establish analogs of several fundamental results known for matroids and linear codes, including the description of minimal supports in terms of cocircuits of truncations and a Greene-type identity relating higher rank-weight enumerators to rank generating functions. We also show that the associated $(q,m)$-polymatroid and the higher support distributions determine each other. As a further application, we derive MacWilliams-type identities for higher rank-weight enumerators.

math.CO↗

arXiv2026

On Representing Matroids via Modular Independence

We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily a matroid. Under a mild nilpotent hypothesis, we show that chain rings are exactly the local rings for which the minimal number of generators is monotone on finitely generated submodules, and over commutative chain rings we obtain a criterion for the associated independence system to be a matroid. For codes over finite commutative chain rings, we identify puncturing with deletion, show that shortening agrees with contraction under a contractibility hypothesis, and establish duality for free codes. We further derive bounds for simple and uniform matroids, prove that the uniform matroid $U_{2, n}$ is representable if and only if the size $n$ is at most the sum of the cardinalities of the local ring and its unique maximal ideal, and show that all excluded minors for $\mathbb{F}_{4}$-representability are representable over $\mathbb{Z}/4\mathbb{Z}$. The examples also include ring representations of matroids not representable over any field, such as the Vámos matroid over $\mathbb{Z}/8\mathbb{Z}$.

math.CO↗

arXiv2025

On the one-dimensional extensions of $q$-matroids

In this paper we introduce a $q$-analogue of the single-element extensions of matroids for $q$-matroids, which we call one-dimensional extensions. To enumerate such extensions, we define a $q$-analogue of modular cuts and define a certain function which we call a modular cut selector. It assigns each newly appearing one-dimensional subspace to a modular cut. By using these notion, we prove the one-to-one correspondence between the one-dimensional extensions and the modular cut selectors. Furthermore, we define the canonnical representatives of the isomorphic class of the $q$-matroids, which enable us to enumerate non-isomorphic $q$-matroids without the paiwise isomorphism testing. As an application, we develop a classification algorithm for $q$-matroids, and classify all the $q$-matroids on ground spaces over $\mathbb{F}_2$ and $\mathbb{F}_3$ of dimension $4$ and $5$ respectively. We also determine some $5$-dimensional $q$-matroids related to the $q$-Fano plane, which is the $q$-analogue of the Fano plane, over $\mathbb{F}_2$.

math.CO↗