Matrices with restricted entries, skew polynomials and coding theory

Coding theory can be seen as the theory of subsets/subspaces of a certain metric space. The most studied and known setting is undoubtedly the Hamming metric. There, one consider a finite dimensional vector space over a (finite) field K, and the distance between two vectors is the number of entries in which they differ. In the last decades, the attention of many researchers has been shifted to rank-metric codes, i.e. linear spaces of matrices over a field K in which the metric considered is given by the rank. Very recently, the sum-rank metric has attracted many people: here the ambient space consists of t-uples of matrices of fixed sizes over a field K, while the distance between two tuples is obtained by adding up the ranks of the differences of the constituent matrices. However, all these metric spaces can be considered as special cases of a more general setting: spaces of matrices with restricted entries, in which the distance of two matrices is the rank of their difference. In this talk I will present a unifying framework for all these metric spaces, which connects them with special spaces of matrices. In particular, I will show how these metric spaces are isometric to suitable quotients of skew polynomial rings, and how to construct linear spaces of matrices with restricted entries in which every nonzero matrix has high rank.