Self-dual codes and LCD codes in sum-rank metric
Qingfeng Xia, Hongwei Liu, Hao Chen, Xu Pan
TL;DR
This work develops a framework for self-dual and LCD codes in the sum-rank metric, leveraging the sum-rank duality theory to enable practical constructions. It introduces two explicit construction methods that lift Euclidean self-dual/LCD codes to the sum-rank setting, producing numerous cyclic examples with strong parameters and demonstrating asymptotic goodness for self-dual sum-rank codes. The results expand the toolkit for sum-rank code design with potential applications in linear network coding, space-time coding, and distributed storage, and also establish the existence of asymptotically good self-dual codes in this metric. Overall, the paper advances both the theory and practice of duality-based code construction in the sum-rank domain.
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.