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 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k]_{q^{m}}$ </tex-math></inline-formula> LRS codes with support-constrained generator matrix is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\geq \ell +1$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq \max _{l\in [\ell]}\{k-1+\log _{q}k, n_{l}\}$ </tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> is the number of blocks and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{l}$ </tex-math></inline-formula> 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.