A General Family of MSRD Codes and PMDS Codes with Smaller Field Sizes from Extended Moore Matrices

We construct six new explicit families of linear maximum sum-rank distance (MSRD) codes, each of which has the smallest field sizes among all known MSRD codes for some parameter regime. Using them and a previous result of the author, we provide two new explicit families of linear partial maximum distance separable (PMDS) codes with smaller field sizes than previous PMDS codes for some parameter regimes. Our approach is to characterize evaluation points that turn extended Moore matrices into the parity-check matrix of a linear MSRD code. We then produce such sequences from codes with good Hamming-metric parameters. The six new families of linear MSRD codes with smaller field sizes are obtained using MDS codes, Hamming codes, Bose--Chaudhuri--Hocquenghem codes, and three algebraic-geometry codes. The MSRD codes based on Hamming codes, of minimum sum-rank distance 3, meet a recent bound by Byrne, Gluesing-Luerssen, and Ravagnani [IEEE Trans. Inform. Theory, 67 (2021), pp. 6456--6475].