A Class of $(n, k, r, t)_i$ LRCs Via Parity Check Matrix

TL;DR

This paper investigates a specific class of locally repairable codes with multiple disjoint repair sets, analyzing their structure via parity check matrices, deriving bounds, and providing constructions for optimal codes using finite field Cayley tables.

Contribution

It introduces a new class of $(n, k, r, t)_i$ LRCs with exactly one parity coordinate per repair set, explores their structural properties, and offers new constructions and bounds.

Findings

Structural features of the parity check matrix are characterized.

Bounds on code parameters are established using matrix-based proofs.

Explicit constructions of optimal codes are provided using finite field Cayley tables.

Abstract

A code is called $(n, k, r, t)$ information symbol locally repairable code \big($(n, k, r, t)_i$ LRC\big) if each information coordinate can be achieved by at least $t$ disjoint repair sets, containing at most $r$ other coordinates. This paper considers a class of $(n, k, r, t)_i$ LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, some structural features of the parity check matrix are proposed by showing some connections with the membership matrix and the minimum distance optimality of the code. Next to that, parity check matrix based proofs of various bounds associated with the code are placed. In addition to this, we provide several constructions of optimal $(n, k, r, t)_i$ LRCs, with the help of two Cayley tables of a finite field. Finally, we generalize a result of $q$-ary $(n, k, r)$ LRCs to $q$-ary $(n, k, r, t)$ LRCs.