New Perfect and Distance-Optimal Sum-Rank Codes

Constructions of infinite families of distance-optimal codes in the Hamming metric and the sum-rank metric are challenging problems and have attracted many attentions. In this paper, we give the following three results. 1) If $\lambda|q^{sm}-1$ and $\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$, an infinite family of distance-optimal $q$-ary cyclic sum-rank codes with the block length $t=\frac{q^{sm}-1}{\lambda}$, the matrix size $s \times s$, 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 $2 \times 2$ 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 $n$ and $m$ satisfying $m<n$, an infinite family of perfect sum-rank codes with the matrix size $m \times n$, and the minimum sum-rank distance three is also constructed. The construction of perfect sum-rank codes of the matrix size $m \times n$, $1<m<n$, answers a problem proposed by U. Mart\'{\i}nez-Pe\~{n}as in 2019 positively. We show that more distance-optimal sum-rank codes can be obtained from the Plotkin sum.