  In this paper, we study vector codes with all-symbol locality, where the
local code is either a Minimum Bandwidth Regenerating (MBR) code or a Minimum
Storage Regenerating (MSR) code. In the first part, we present vector codes
with all-symbol MBR locality, for all parameters, that have both optimal
minimum-distance and optimal rate. These codes combine ideas from two popular
codes in the distributed storage literature, Product-Matrix codes and Tamo-Barg
codes. In the second part which deals with codes having all-symbol MSR
locality, we follow a Pairwise Coupling Transform-based approach to arrive at
optimal minimum-distance and optimal rate, for a range of parameters. All the
code constructions presented in this paper have a low field-size that grows
linearly with the code-length $n$.
