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

TL;DR

This paper characterizes the existence conditions for support-constrained Linearized Reed-Solomon codes, which are optimal in the sum-rank metric, and explores their applications in multi-source network coding.

Contribution

It provides necessary and sufficient conditions for support-constrained MSRD LRS codes and links these conditions to network coding applications.

Findings

Support constraints mirror those of MDS and MRD codes.

Field size requirements for constructing such codes are established.

Applications include designing distributed LRS codes via integer programming.

Abstract

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.