Long Optimal or Small-Defect LRC Codes with Unbounded Minimum Distances

TL;DR

This paper constructs new optimal locally recoverable codes (LRCs) over finite fields with lengths around the square of the field size, unbounded localities, and large minimum distances, filling a significant gap in coding theory.

Contribution

It introduces a family of optimal LRCs with lengths around the square of the field size, unbounded localities, and minimum distances at least 9, which was previously unachieved.

Findings

Constructed optimal LRCs with length around q^2, locality r, and minimum distance d.

Provided an asymptotic bound for q-ary r-locality LRC codes surpassing previous bounds.

Developed long LRCs with small defects for practical applications.

Abstract

A code over a finite field is called locally recoverable code (LRC) if every coordinate symbol can be determined by a small number (at most r, this parameter is called locality) of other coordinate symbols. For a linear code with length n, dimension k and locality r, its minimum distance d satisfies a Singleton-like bound. A code attaining this bound is called optimal. Many families of optimal locally recoverable codes have been constructed by using different techniques in finite fields or algebraic curves. However no optimal LRC code over a general finite field of q elements with the length n around the square of q, the locality r larger than or equal to 24 and the minimum distance d larger than or equal to 9 has been constructed. In this paper for any given finite field of q elements, any given r between 1 and q-1 and given d in certain range, we give an optimal LRC code with length n around the square of q, locality r and minimum distance d. This is the only known family of optimal LRC codes with lengths n around the square of q and unbounded localities and minimum distances d larger than or equal to 9. We also gives an asymptotic bound for q-ary r locality LRC codes better than the previously konwn bound. Many long r-locality LRC codes with small defects are also constructed.