Some Matroids Related to Sum-Rank Metric Codes

Panja, Avijit; Pratihar, Rakhi; Randrianarisoa, Tovohery Hajatiana

doi:10.48550/arxiv.1912.09984

Cited by 2 publications

(4 citation statements)

References 26 publications

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“…Hence, an application of Theorem 3.2 yields a Wei-type duality theorem for q-matroid. This result generalizes the Wei-type duality theorems for matroids [6,Theorem 1], for sum-matroids [32,Theorem 12] and for sum-rank metric codes [27,Theorem 2]. 4 Wei-type duality theorems for w-demimatroids Throughout this section, we let E be a finite set with |E| = m. We begin with the definition of w-demimatroid.…”

Section: The Second Bridging Theoremmentioning

confidence: 66%

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A Galois Connection Approach to Wei-Type Duality Theorems

Xu¹,

Kan²,

Han³

2020

Preprint

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In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei's duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei's duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result in this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of Z, from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for w-demimatroids defined over finite sets and w-demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.

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Section: The Second Bridging Theoremmentioning

confidence: 66%

“…A similar approach to [17] for demi-polymatroids was independently proposed by Britz, Mammomiti and Shiromoto in [7]. Recently, Panja, Pratihar and Hajatiana Randrianarisoa proved [32] a Wei-type duality theorem for sum-matroids, which is another generalization of the q-analogue of a matroid.…”

Section: Introductionmentioning

confidence: 89%

A Galois Connection Approach to Wei-Type Duality Theorems

Xu¹,

Kan²,

Han³

2020

Preprint

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“…After the reintroduction of the object by Jurrius and Pellikaan [10] and independently that of (q, r)-polymatroids by Shiromoto [14], several other papers have studied these objects, often in relation to rank metric codes. See for example [2,4,5,6,7,9,12].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we consider the direct sum of two q-matroids. An option to do this is to extend to the realm of sum-matroids [12], but we are looking for a construction that gives a q-matroid. This is one of the cases as mentioned above where the q-analogue is a lot harder than the relatively simple procedure of taking the direct sum of two classical matroids.…”

Section: Introductionmentioning

confidence: 99%

The direct sum of $q$-matroids

Ceria¹,

Jurrius²

2021

Preprint

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For classical matroid, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of q-polymatroids and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.

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Some Matroids Related to Sum-Rank Metric Codes

Cited by 2 publications

References 26 publications

A Galois Connection Approach to Wei-Type Duality Theorems

A Galois Connection Approach to Wei-Type Duality Theorems

The direct sum of $q$-matroids

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