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