Concatenated Sum-Rank Codes

TL;DR

This paper introduces a concatenation method combining sum-rank and Hamming metric codes, leading to explicit constructions of superior sum-rank codes that outperform existing bounds and previous codes.

Contribution

It presents a novel concatenation technique for sum-rank codes with Hamming codes, resulting in explicit, better-performing codes and an asymptotically good sequence surpassing known bounds.

Findings

Constructed many sum-rank codes better than sum-rank BCH codes.

Developed an explicit concatenation method for sum-rank and Hamming codes.

Produced an asymptotically good sequence exceeding established bounds.

Abstract

Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the Gilbert-Varshamov-like bound, were constructed in a recent paper published in IEEE Trans. Inf. Theory by E. Berardini and X. Caruso. We call this bound the Tsfasman-Vladut-Zink-like bound. In this paper, we introduce the concatenation of a sum-rank code and a Hamming metric code. Then many sum-rank codes with good parameters, which are better than sum-rank BCH codes, are constructed simply and explicitly. Moreover, we obtain an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vladut-Zink-like bound and the Gilbert-Varshamov-like bound.