Optimal quaternary $(r,\delta)$-Locally Recoverable Codes: Their Structures and Complete Classification

TL;DR

This paper provides a complete structural classification of optimal quaternary $(r, \, \delta)$-locally recoverable codes over the field \(\mathbb{F}_4\), using parity-check matrices, advancing understanding of their parameters and constructions.

Contribution

It offers the first complete classification of optimal quaternary $(r, \, \delta)$-LRC codes based on their parity-check matrix structures.

Findings

Structural properties of optimal codes identified

Explicit constructions of optimal codes provided

Complete classification over \(\mathbb{F}_4\) achieved

Abstract

Aiming to recover the data from several concurrent node failures, linear $r$-LRC codes with locality $r$ were extended into $(r, \delta)$-LRC codes with locality $(r, \delta)$ which can enable the local recovery of a failed node in case of more than one node failure. Optimal LRC codes are those whose parameters achieve the generalized Singleton bound with equality. In the present paper, we are interested in studying optimal LRC codes over small fields and, more precisely, over $\mathbb{F}_4$. We shall adopt an approach by investigating optimal quaternary $(r,\delta)$-LRC codes through their parity-check matrices. Our study includes determining the structural properties of optimal $(r,\delta)$-LRC codes, their constructions, and their complete classification over $\F_4$ by browsing all possible parameters. We emphasize that the precise structure of optimal quaternary $(r,\delta)$-LRC codes and their classification are obtained via the parity-check matrix approach use proofs-techniques different from those used recently for optimal binary and ternary $(r,\delta)$-LRC codes obtained by Hao et al. in [IEEE Trans. Inf. Theory, 2020, 66(12): 7465-7474].