Self-dual codes and LCD codes in sum-rank metric

TL;DR

This paper explores the theory and construction of self-dual and LCD codes within the sum-rank metric, providing new methods, examples, and asymptotic existence results relevant for network coding and data storage.

Contribution

It introduces the notions of self-dual and LCD codes in sum-rank metric and offers new construction methods from Euclidean codes, along with examples and asymptotic existence proofs.

Findings

Constructed new classes of self-dual sum-rank codes

Constructed LCD sum-rank codes with good parameters

Proved the existence of asymptotically good self-dual sum-rank codes

Abstract

Sum-rank codes are an important class of codes which can be utilized for linear network coding, space-time coding and distributed storage. They can not only reduce the size of network alphabet but also detect and correct more errors. Based on the duality theory of sum-rank codes [Byrne, Gluesing-Luerssen, Ravagnani, IEEE TIT, 2021] and those related theory of rank-metric codes, it is significant to study self-dual codes and linear complementary dual (LCD) codes in sum-rank metric. In this paper, we introduce the notion of self-dual codes and LCD codes in sum-rank metric, and obtain two methods of constructing self-dual sum-rank codes and LCD sum-rank codes from Euclidean self-dual codes and Euclidean LCD codes. Some examples of cyclic self-dual sum-rank codes and cyclic LCD sum-rank codes with good parameters are provided. In addition, we prove that there exist asymptotically good self-dual sum-rank codes.