Matroids and Codes with the Rank Metric
Keisuke Shiromoto
TL;DR
This paper explores the connection between q-analogue matroids and rank metric codes, establishing identities that link their combinatorial and algebraic properties, with implications for coding theory.
Contribution
It introduces a Greene type identity for rank generating functions and provides a combinatorial proof of a MacWilliams identity for Delsarte rank-metric codes.
Findings
Proved a Greene type identity for rank generating functions.
Established a MacWilliams type identity for Delsarte rank-metric codes.
Linked matroid theory with rank metric code properties.
Abstract
We study the relationship between a q-analogue of matroids and linear codes with the rank metric in the vector space of matrices with entries in a finite field. We prove a Greene type identity for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes. As an application, we give a combinatorial proof of a MacWilliams type identity for Delsarte rank-metric codes.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research