2022/12/15 by Hedongliang Liu, Liu, Hedongliang, Hengjia Wei +5 · 1 citation
Computer Science · Engineering · #Advanced Wireless Communication Technologies #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.2212.07991
openalex publication_date 2022/12/15 · openalex created_date 2022/12/28 · openalex updated_date 2026/07/28
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]qm LRS codes with support-constrained generator matrix is q≥ ℓ+1 and m≥ maxl∈[ℓ]\k-1+logqk, nl\, where ℓ is the number of blocks and nl 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.