Linearized Reed-Solomon Codes with Support-Constrained Generator Matrix
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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 [n,k]_q^m LRS codes with support-constrained generator matrix is q≥ℓ+1 and m≥max_l∈[ℓ]{k-1+log_qk, n_l}, where ℓ 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.
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