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 $\lambda|q^{sm}-1$ 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 $m$ satisfying $2 \leq m$, an infinite family of quasi-perfect sum-rank codes with the matrix size $2 \times m$, 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.

翻译：距离最优码与准完备码的构造是具有挑战性的问题，并吸引了广泛关注。本文给出以下三个结果：1) 当 $\lambda|q^{sm}-1$ 且 $\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$ 时，构造了一族无限类距离最优 $q$ 元循环求和秩码，其块长 $t=\frac{q^{sm}-1}{\lambda}$，矩阵尺寸 $s \times s$，基数 $q^{s^2t-s(2m+3)}$，最小求和秩距离为4。2) 构造了块长为 $q^4-1$、矩阵尺寸 $2 \times 2$ 的距离最优求和秩码，其最小求和秩距离为4，Singleton 缺陷为4。这些求和秩码逼近球堆积界和类Singleton界，且具有显著更长的块长 $q^4-1>>q-1$。3) 对于满足 $2 \leq m$ 的给定正整数 $m$，构造了一族无限类准完备求和秩码，矩阵尺寸 $2 \times m$，最小求和秩距离为3。同时给出了最小求和秩距离为4的准完备二元求和秩码。还给出了块长可达 $q^2$ 的几乎MSRD $q$ 元码。我们证明通过Plotkin和可得到更多距离最优二元求和秩码。