论文标题
新的显式线性总和级金属代码
New Explicit Good Linear Sum-Rank-Metric Codes
论文作者
论文摘要
总和 - 金属代码在通用误差校正，多疗法网络编码，时空编码以及分布式存储中维修的部分MDS代码中具有广泛的应用。已经研究了总和 - 金属代码的基本属性，并提出了良好总和 - 金属代码的一些明确或概率构造。在本文中，我们提供了三个简单的明确线性总和级 - 金属代码的结构。在有限的长度制度中，可以从我们的构造中得出许多与先前构造的代码相同的最小总和率距离的较大较大的线性总和级代码。例如，几个更好的线性总和级 - 金属代码超过$ {\ bf f} _q $，尺寸很小，矩阵尺寸$ 2 \ times 2 $由$ q = 2、3、4 $构建，通过将我们的构造应用于目前已知的最佳最佳线性码。渐近地，我们构造的总和 - 金属代码接近某些参数的吉尔伯特·瓦尔沙莫夫（Gilbert-Varshamov）类似于总和 - 金属代码的绑定。最后，我们在任意有限字段$ {\ bf f} _q $上构造线性MSRD代码，具有各种平方矩阵$ n_1，n_2，n_2，\ ldots，n_t $满足$ n_i \ geq n_ \ geq n_ \ geq n_ {i+1} 距离。这些线性MSRD代码的块长度$ t $和参数没有限制，$ n = n_1+\ cdots+n_t $从字段$ {\ bf f} _q $的尺寸中。 \ end {摘要}
Sum-rank-metric codes have wide applications in universal error correction, multishot network coding, space-time coding and the construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed. In this paper we give three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous larger linear sum-rank-metric codes with the same minimum sum-rank distances as the previous constructed codes can be derived from our constructions. For example several better linear sum-rank-metric codes over ${\bf F}_q$ with small block sizes and the matrix size $2 \times 2$ are constructed for $q=2, 3, 4$ by applying our construction to the presently known best linear codes. Asymptotically our constructed sum-rank-metric codes are close to the Gilbert-Varshamov-like bound on sum-rank-metric codes for some parameters. Finally we construct a linear MSRD code over an arbitrary finite field ${\bf F}_q$ with various square matrix sizes $n_1, n_2, \ldots, n_t$ satisfying $n_i \geq n_{i+1}^2+\cdots+n_t^2$ , $i=1, 2, \ldots, t-1$, for any given minimum sum-rank distance. There is no restriction on the block lengths $t$ and parameters $N=n_1+\cdots+n_t$ of these linear MSRD codes from the sizes of the fields ${\bf F}_q$. \end{abstract}
