A general family of MSRD codes and PMDS codes with smaller field sizes from extended Moore matrices

Umberto Martínez-Peñas

In this work, a general method for constructing linear maximum sum-rank distance (MSRD) codes is introduced. By a previous result of the author, any of these MSRD codes provides a linear partial-MDS (PMDS) code. For the MSRD code constructions, extended Moore matrices are introduced. These matrices extend generator matrices of linearized Reed-Solomon codes, in the sense that evaluation points inside a conjugacy class need not be linearly independent over the base field. The key result of this work is a characterization of evaluation points per conjugacy class that turn extended Moore matrices into the parity-check (or generator) matrix of a linear MSRD code. The sufficient and necessary conditions on the evaluation points constitute a natural generalization of the geometric concept of (partial) spread. Extending Segre's original construction of spreads, we provide a method based on tensor products to produce satisfactory sequences of evaluation points. The method takes as input a Hamming-metric code and gives as output a linear MSRD code. A list of linear MSRD codes admitting a wide range of parameters is then obtained, giving as input trivial codes (yielding linearized Reed-Solomon codes), MDS codes, Hamming codes, BCH codes and several Algebraic-Geometry codes. Each of the obtained MSRD codes attains the smallest known field size, or the largest number of matrix sets, for some parameter regime. In particular, the MSRD codes based on Hamming codes, valid for minimum sum-rank distance $ 3 $, meet a recent bound by Byrne et al. These codes are also the first and only known MSRD codes with field sizes that are linear in the code length if the number of columns per matrix is constant. Finally, two new families of PMDS codes are obtained attaining smaller field sizes than those in the literature for many parameter regimes.

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