  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 MDS (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, BCH 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 et al.
