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 the open problem proposed by U. Mart\'{\i}nez-Pe\~{n}as in 2019 positively.

翻译：在汉明度量与和秩度量下构造无限族距离最优码是具有挑战性的问题，并引起了广泛关注。本文给出以下三个结果：1) 若 $\lambda|q^{sm}-1$ 且 $\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$，构造了块长 $t=\frac{q^{sm}-1}{\lambda}$、矩阵尺寸 $s \times s$、基数 $q^{s^2t-s(2m+3)}$ 且最小和秩距离为4的无限族距离最优 $q$ 元循环和秩码。2) 构造了块长 $q^4-1$、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。这些和秩码接近球填充界和Singleton类界，且具有更大的块长 $q^4-1>>q-1$。3) 对满足 $m<n$ 的正整数 $n$ 和 $m$，构造了矩阵尺寸 $m \times n$、最小和秩距离为3的无限族完美和秩码。矩阵尺寸 $m \times n$（$1<m<n$）的完美和秩码的构造，正面解答了2019年由U. Martínez-Peñas提出的开放问题。