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.

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，并吸引了广泛关注。本文取得以下三项成果：1) 若$\lambda|q^{sm}-1$且$\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$，构造了一类无限族距离最优的$q$-元循环和秩码，其块长$t=\frac{q^{sm}-1}{\lambda}$，矩阵尺寸$s \times s$，基数$q^{s^2t-s(2m+3)}$，最小和秩距离为4；2) 构造了块长$q^4-1$、矩阵尺寸$2 \times 2$、最小和秩距离为4且Singleton缺陷为4的距离最优和秩码。这些和秩码逼近球堆积界与Singleton型界，且具有远大于$q-1$的块长$q^4-1$；3) 对于满足$m<n$的给定正整数$n$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。该构造肯定地回答了U. Martíne-Peñas于2019年提出的关于$1<m<n$矩阵尺寸完美和秩码的开放问题。我们进一步证明，通过Plotkin和运算可得到更多距离最优和秩码。