Linearized Reed-Solomon (LRS) codes are a class of evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric, and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions on the existence of MSRD codes with support-constrained generator matrix. These conditions are identical to those for MDS codes and MRD codes. Moreover, the required field size for an $[n,k]_{q^m}$ LRS codes with support-constrained generator matrix is $q\geq \ell+1$ and $m\geq \max_{l\in[\ell]}\{k-1+\log_qk, n_l\}$, where $\ell$ is the number of blocks and $n_l$ is the size of the $l$-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes.
翻译：线性Reed-Solomon (LRS) 代码是一类基于 skew 多元调制的评审代码。 它们实现了单吨约束值, 以总和量度计, 并因此被称为最大总和距离值( MSRD) 代码。 在这项工作中, 我们为MSRD 代码的存在提供了必要和足够的条件, 并配有支持限制的生成器矩阵。 这些条件与 MDS 代码和 MRD 代码的条件相同。 此外, $( k) 和 $( Q) 的LRS 代码与支持限制的生成器矩阵所需的字段大小是 $q\geq =ell+1$ 和 $m\geq\\ max ⁇ l\ l\ in[ \\\\\\\\\ k\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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