Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes

Constructions of distance-optimal codes and quasi-perfect codes are challenging problems and have attracted many attentions. In this paper, we give the following three results. 1) If λ|q^sm-1 and λ <√((q^s-1)/2(q-1)^2(1+ϵ)), an infinite family of distance-optimal q-ary cyclic sum-rank codes with the block length t=q^sm-1/λ, the matrix size s × s, the cardinality q^s^2t-s(2m+3) and the minimum sum-rank distance four is constructed. 2) Block length q^4-1 and the matrix size 2 × 2 distance-optimal sum-rank codes with the minimum sum-rank distance four and the Singleton defect four are constructed. These sum-rank codes are close to the sphere packing bound , the Singleton-like bound and have much larger block length q^4-1>>q-1. 3) For given positive integers m satisfying 2 ≤ m, an infinite family of quasi-perfect sum-rank codes with the matrix size 2 × m, and the minimum sum-rank distance three is also constructed. Quasi-perfect binary sum-rank codes with the minimum sum-rank distance four are also given. Almost MSRD q-ary codes with the block lengths up to q^2 are given. We show that more distance-optimal binary sum-rank codes can be obtained from the Plotkin sum.