Constructions and List Decoding of Sum-Rank Metric Codes Based on Orthogonal Spaces over Finite Fields

TL;DR

This paper introduces new sum-rank metric codes based on orthogonal spaces over finite fields, constructs specific MRD codes, and analyzes list decoding sizes to enhance decoding success in network and storage applications.

Contribution

It presents novel constructions of sum-rank metric codes using orthogonal groups and subspace designs, along with methods to calculate list decoding sizes, improving decoding success rates.

Findings

Constructed cyclic and Abelian orthogonal groups over finite fields.

Developed two methods for constructing sum-rank metric codes from MRD codes.

Calculated list sizes under list decoding, showing improved success rates.

Abstract

Sum-rank metric codes, as a generalization of Hamming codes and rank metric codes, have important applications in fields such as multi-shot linear network coding, space-time coding and distributed storage systems. The purpose of this study is to construct sum-rank metric codes based on orthogonal spaces over finite fields, and calculate the list sizes outputted by different decoding algorithms. The following achievements have been obtained. In this study, we construct a cyclic orthogonal group of order $q^n-1$ and an Abelian non-cyclic orthogonal group of order $(q^n-1)^2$ based on the companion matrices of primitive polynomials over finite fields. By selecting different subspace generating matrices, maximum rank distance (MRD) codes with parameters $(n \times {2n}, q^{2n}, n)_q$ and $(n \times {4n}, q^{4n}, n)_q$ are constructed respectively. Two methods for constructing sum-rank metric codes are proposed for the constructed MRD codes, and the list sizes outputted under the list decoding algorithm are calculated. Subsequently, the $[{\bf{n}},k,d]_{{q^n}/q}$-system is used to relate sum-rank metric codes to subspace designs. The list size of sum-rank metric codes under the list decoding algorithm is calculated based on subspace designs. This calculation method improves the decoding success rate compared with traditional methods.