Linearized Reed-Solomon Codes with Support-Constrained Generator Matrix
Hedongliang Liu, Hengjia Wei, Antonia Wachter-Zeh, Moshe Schwartz · 2023
Linearized Reed-Solomon (LRS) codes are a class of 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 on the existence of MSRD codes with support-constrained generator matrix. These conditions are identical to those for MDS codes and MRD codes. Moreover, the required field size for an ${\left[ {n,k} \right]_{{q^m}}}$ 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 nlis 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.