Linearized Reed-Solomon (LRS) codes are 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 for the existence of MSRD codes with a support-constrained generator matrix. The conditions on the support constraints are identical to those for MDS codes and MRD codes. 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. For the support constraints that do not satisfy the necessary conditions, we derive the maximum sum-rank distance of a code whose generator matrix fulfills the constraints. Such a code can be constructed from a subcode of an LRS code with a sufficiently large field size. Moreover, as an application in network coding, the conditions can be used as constraints in an integer programming problem to design distributed LRS codes for a distributed multi-source network.

翻译：线性化Reed-Solomon（LRS）码是基于斜多项式的求值码。它们在和秩度量下达到Singleton界，因此被称为最大和秩距离（MSRD）码。本文给出了具有支撑约束生成矩阵的MSRD码存在的充分必要条件。该支撑约束条件与MDS码和MRD码的约束条件相同。对于具有支撑约束生成矩阵的$[n,k]_{q^m}$ LRS码，所需的域大小为$q\geq \ell+1$且$m\geq \max_{l\in[\ell]}\{k-1+\log_qk, n_l\}$，其中$\ell$为分块数，$n_l$为第$l$个分块的大小。该结论的特例与已知的Reed-Solomon码和Gabidulin码的结果一致。对于不满足必要条件的支撑约束，我们推导了其生成矩阵满足该约束的码的最大和秩距离。此类码可通过具有足够大域尺寸的LRS码的子码构造。此外，作为网络编码中的一个应用，这些条件可作为整数规划问题的约束，用于设计分布式多源网络的分布式LRS码。