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

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 $n$.