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. 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，并已引起广泛关注。本文给出以下三项结果：1) 当$\lambda|q^{sm}-1$且$\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$时，构造了一类无穷族距离最优$q$元循环和秩码，其分块长度$t=\frac{q^{sm}-1}{\lambda}$、矩阵尺寸$s \times s$、基数$q^{s^2t-s(2m+3)}$且最小和秩距离为4。2) 构造了分块长度$q^4-1$、矩阵尺寸$2 \times 2$且最小和秩距离为4、Singleton缺陷为4的距离最优和秩码。这些和秩码逼近球堆积界和Singleton类似界，且分块长度$q^4-1>>q-1$更大。3) 对于满足$2 \leq m$的给定正整数$m$，同时构造了一类无穷族准完美和秩码，其矩阵尺寸$2 \times m$、最小和秩距离为3。我们进一步证明，通过Plotkin和运算可获得更多距离最优二元和秩码。