Theory of supports for linear codes endowed with the sum-rank metric

Abstract

The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (support spaces) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: 1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; 2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; 3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.

The geometry of one-weight codes in the sum-rank metric

“…In the last few years, a deep mathematical theory of sum-rank-metric codes was developed in a series of papers by Martínez-Peñas [26,29,30]. These codes can be seen as a generalization of Hamming-metric codes and rank-metric codes.…”

“…We recall a slightly different notion of sum-rank metric code, in which the codewords are vectors with entries from an extension field F q m rather than matrices over F q . The interested reader is referred to [26,28,29,31,34] for a more detailed description of this setting. The F q -rank of a vector v = (v 1 , .…”

“…Hence, we can give the following definitions of sum-rank support, rank-list and rank-profile for elements in F n q m . For a deeper understanding of these notions we refer the reader to [26]. Definition 2.10.…”

A new rank metric for convolutional codes

“…The sum rank distance, the distance that has been widely considered for multishot network coding, was first introduced in [27] under the name of extended rank distance. We note that the sum rank distance has been also used in the context of block codes reducing their decoding complexity, see for instance [21].…”

“…Thus, some sequences are not considered in the time interval [0, j] and therefore this metric is not a sufficient metric to guarantee decoding within a time interval. Hence, a new distance, called sum rank distance, was introduced as a generalization of the active column rank distance and the rank distance used for one-shot network coding (see [19], [21] and [27]). This new distance has proven to be the proper notion in order to deal with networks that are delay-free.…”