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

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.