A sum-rank-metric code attaining the Singleton bound is called maximum sum-rank distance (MSRD). MSRD codes have applications in space-time coding and construction of partial-MDS codes for repair in distributed storage. MSRD codes have been constructed in some parameter cases. In this paper we construct a ${\bf F}_q$-linear MSRD code over some field ${\bf F}_q$ with different matrix sizes $n_1>n_2>\cdots>n_t$ satisfying $n_i \geq n_{i+1}^2+\cdots+n_t^2$ for $i=1, 2, \ldots, t-1$ for any given minimum sum-rank distance. Many good linear sum-rank-metric codes over small fields with such different matrix sizes are given. A lower bound on the dimensions of constructed ${\bf F}_{q^2}$-linear sum-rank-metric codes over ${\bf F}_{q^2}$ with such different matrix sizes and given minimum sum-rank distances is also presented.
翻译：达到单吨约束值的平面代码称为最大平面距离(MSRD)。MSRD代码可用于空间时间编码和在分布式储存中建造部分MDS代码。MSRD代码在一些参数情况下已经建成。在本文中,我们在一些字段中建造了美元=bf F ⁇ q$-线性MSRD代码,其基号不同,其基号为$_1>n_2 ⁇ cdots>n_t$满足 $_i\geq n ⁇ 2 ⁇ cdos+n_t%2$,用于美元=1,2\ldots,对于任何给定的最低平面距离为t-1美元。还给出了许多对地表大小不同的小田的精细线性平面码。还给出了在建基号为$_bf F ⁇ 2}(美元=f f ⁇ q ⁇ 2}(美元=f f ⁇ q ⁇ 2}的线性平面线性平面代码尺寸较低,其尺寸不同,其基号尺寸不同,并给予最小的平面距离。
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