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

TL;DR

This paper introduces new vector codes with all-symbol locality that achieve optimal minimum-distance and rate, combining ideas from MBR and MSR codes with low field-size growth, enhancing distributed storage efficiency.

Contribution

The paper presents novel vector codes with all-symbol MBR and MSR locality that are optimal in minimum-distance and rate, using low field-size constructions.

Findings

Codes with all-symbol MBR locality are optimal in minimum-distance and rate.

Codes with all-symbol MSR locality are also optimal with a pairwise coupling transform.

All constructions have low, linearly growing field-size.

Abstract

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 $n$.