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：距离最优码和准完美码的构造是极具挑战性的问题，长期以来备受关注。本文给出以下三个主要结果：1) 当 $\lambda|q^{sm}-1$ 且 $\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$ 时，构造了一个无限族距离最优的 $q$ 元循环和秩码，其分组长度 $t=\frac{q^{sm}-1}{\lambda}$，矩阵尺寸 $s \times s$，基数 $q^{s^2t-s(2m+3)}$，最小和秩距离为4。2) 构造了分组长度为 $q^4-1$、矩阵尺寸为 $2 \times 2$ 的距离最优和秩码，其最小和秩距离为4，Singleton缺损为4。这些和秩码接近球包界和Singleton型界，且具有显著更大的分组长度 $q^4-1>>q-1$。3) 对于满足 $2 \leq m$ 的给定正整数 $m$，构造了一个无限族准完美和秩码，矩阵尺寸为 $2 \times m$，最小和秩距离为3。同时给出了最小和秩距离为4的准完美二元和秩码。进一步构造了分组长度可达 $q^2$ 的几乎MSRD $q$ 元码。研究表明，通过Plotkin和运算可得到更多距离最优二元和秩码。