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

Codes with Combined Locality and Regeneration Having Optimal Rate, $d_{\text{min}}$ and Linear Field Size