Linearized Reed-Solomon Codes with Support-Constrained Generator Matrix and Applications in Multi-Source Network Coding

Linearized Reed-Solomon (LRS) codes are evaluation codes based on skewpolynomials. They achieve the Singleton bound in the sum-rank metric andtherefore are known as maximum sum-rank distance (MSRD) codes. In this work, wegive necessary and sufficient conditions for the existence of MSRD codes with asupport-constrained generator matrix. These conditions are identical to thosefor MDS codes and MRD codes. The required field size for an $[n,k]_{q^m}$ LRScodes 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 blocksand $n_l$ is the size of the $l$-th block. The special cases of the resultcoincide with the known results for Reed-Solomon codes and Gabidulin codes. Forthe support constraints that do not satisfy the necessary conditions, we derivethe maximum sum-rank distance of a code whose generator matrix fulfills theconstraints. Such a code can be constructed from a subcode of an LRS code witha sufficiently large field size. Moreover, as an application in network coding,the conditions can be used as constraints in an integer programming problem todesign distributed LRS codes for a distributed multi-source network.