Shellability and homology of $q$-complexes and $q$-matroids

Ghorpade, Sudhir R.; Pratihar, Rakhi; Randrianarisoa, Tovohery Hajatiana

doi:10.48550/arxiv.2102.13102

Cited by 1 publication

(1 citation statement)

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“…Following the work on the connection between generalized weights of a Hamming metric code C and Betti numbers of certain associated matroid M as established in [18], an expression for A C,j (Q) or more generally, for the generalized weight polymials P M,j (Q) of a matroid M, is provided in terms of Betti numbers associated to the Stanley-Reisner ring of the matroid M and its elongations. On the other hand, recently a study on determining the singular homology of q-complexes associated to q-matroids has been initiated in [10]. This work is towards a topological approach to connect the generalized rank weights of a rank metric code with homological invariants of the associated q-matroid.…”

Section: Introductionmentioning

confidence: 99%

Weight spectra of Gabidulin rank-metric codes and Betti numbers

Johnsen

Pratihar

Verdure

2022

São Paulo J. Math. Sci.

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We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the q-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for q-matroids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated q-matroids by using Möbius functions of its lattice of q-flats.

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Section: Introductionmentioning

confidence: 99%