# q-Polymatroids and their application to rank-metric codes.

## Author ORCID Identifier

https://orcid.org/0000-0002-4273-5668

5-1-2023

2023

## Degree Name

Doctor of Philosophy (PhD)

## Document Type

Doctoral Dissertation

## College

Arts and Sciences

Mathematics

## Abstract

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and q-matroids. We then describe how linear rank metric codes induce q-polymatroids and show how some invariants of rank-metric codes are fully determined by the induced q-polymatroid. Furthermore, we show that not all q-polymatroids arise from rank metric codes which gives rise to the class of non-representable q-polymatroids. We then define the notion of independent space for q-polymatroids and show that together with their rank values, those independents spaces fully determine the q-polymatroid. Next, we restrict ourselves to the study of q-matroids. We start by studying the characteristic polynomial of q-matroids by relating it to the characteristic polynomial of the projectivazition matroid. We establish a deletion/contraction formula for the characteristic polynomial of q-matroids and prove a q-analogue of the Critical Theorem. Afterwards, we study the direct-sum of q-matroids. We show the cyclic flats of the direct sum can be nicely characterized in terms of the cyclic flats of each summands. Using this characterization, we show all q-matroids can be uniquely decomposed (up to equivalence) into the direct sum of irreducible components. We furthermore show that unlike classical matroids, the direct sum of two representable q-matroids over some fixed field is not necessarily representable over that same field. Finally we consider q-matroids from a category theory perspective to study the theoretical similarities and differences between classical matroids and q-matroids. We define several type of maps between q-matroids and consider the resultant categories. We then proceed to show that the direct sum of q-matroids is a coproduct in only one of those categories which stands in contrast to categories of classical matroids. We conclude by showing the existence of a functor from categories of q-matroids to categories of matroids which provide an alternative method to study the former categories.

## Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2023.131

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