New Explicit Good Linear Sum-Rank-Metric Codes

TL;DR

This paper introduces three explicit constructions of linear sum-rank-metric codes, achieving near-optimal parameters and including a new MSRD code over arbitrary finite fields with flexible matrix sizes.

Contribution

The paper presents three simple, explicit constructions of linear sum-rank-metric codes, including a novel MSRD code applicable over any finite field with flexible parameters.

Findings

Constructed larger linear sum-rank-metric codes with the same minimum distance.

Developed better codes over small fields with small block sizes.

Created a flexible MSRD code over arbitrary finite fields.

Abstract

Sum-rank-metric codes have wide applications in universal error correction, multishot network coding, space-time coding and the construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed. In this paper we give three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous larger linear sum-rank-metric codes with the same minimum sum-rank distances as the previous constructed codes can be derived from our constructions. For example several better linear sum-rank-metric codes over ${\bf F}_q$ with small block sizes and the matrix size $2 \times 2$ are constructed for $q=2, 3, 4$ by applying our construction to the presently known best linear codes. Asymptotically our constructed sum-rank-metric codes are close to the Gilbert-Varshamov-like bound on sum-rank-metric codes for some parameters. Finally we construct a linear MSRD code over an arbitrary finite field ${\bf F}_q$ with various square matrix sizes $n_1, n_2, \ldots, n_t$ satisfying $n_i \geq n_{i+1}^2+\cdots+n_t^2$ , $i=1, 2, \ldots, t-1$, for any given minimum sum-rank distance. There is no restriction on the block lengths $t$ and parameters $N=n_1+\cdots+n_t$ of these linear MSRD codes from the sizes of the fields ${\bf F}_q$. \end{abstract}