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

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

The Projectivization Matroid of a qq-Matroid

In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-matroid, and considering the projective space as the groundset of the associated matroid, on which is defined a rank function compatible with that of the $q$-matroid. We show that the projectivization map is a functor from categories of $q$-matroids to categories of matroids. This relation is used to prove new results about maps of $q$-matroids. Furthermore, we show the characteristic polynomial of a $q$-matroid is equal to that of the projectivization matroid, which we use to establish a recursive formula for the characteristic polynomial of a $q$-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a $q$-analogue of the critical theorem in terms of $\mathbb{F}_{q^m}$-linear rank metric codes and $q$-matroids

A Polymatroid Approach to Generalized Weights of Rank Metric Codes

We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio

Relations between M\"obius and coboundary polynomial

It is known that, in general, the coboundary polynomial and the M\"obius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will try to answer if it is possible that the M\"obius polynomial of a matroid, together with the M\"obius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is affirmative, and we will give two constructions to determine the coboundary polynomial in these cases.Comment: 12 page

On the existence of asymptotically good linear codes in minor-closed classes

Let $\mathcal{C} = (C_1, C_2, \ldots)$ be a sequence of codes such that each $C_i$ is a linear $[n_i,k_i,d_i]$-code over some fixed finite field $\mathbb{F}$, where $n_i$ is the length of the codewords, $k_i$ is the dimension, and $d_i$ is the minimum distance. We say that $\mathcal{C}$ is asymptotically good if, for some $\varepsilon > 0$ and for all $i$, $n_i \geq i$, $k_i/n_i \geq \varepsilon$, and $d_i/n_i \geq \varepsilon$. Sequences of asymptotically good codes exist. We prove that if $\mathcal{C}$ is a class of GF$(p^n)$-linear codes (where $p$ is prime and $n \geq 1$), closed under puncturing and shortening, and if $\mathcal{C}$ contains an asymptotically good sequence, then $\mathcal{C}$ must contain all GF$(p)$-linear codes. Our proof relies on a powerful new result from matroid structure theory

On the Combinatorics of Locally Repairable Codes via Matroid Theory

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters $(n,k,d,r,\delta)$ of LRCs are generalized to matroids, and the matroid analogue of the generalized Singleton bound in [P. Gopalan et al., "On the locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes, that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters $(n,k,d,r,\delta)$ and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given in [W. Song et al., "Optimal locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has been edited to improve the readability. Parameter d for matroids is now defined by the use of the rank function instead of the dual matroid. Typos are corrected. Section III is divided into two parts, and some numberings of theorems etc. have been change