Locally Repairable Codes with Availability via Elliptic Function Fields

TL;DR

This paper introduces new constructions of optimal locally repairable codes using elliptic function fields, broadening the available methods and providing codes with flexible locality and multiple recovering sets.

Contribution

It presents a novel framework for constructing locally repairable codes with two recovering sets via automorphism groups of elliptic function fields, using both supersingular and ordinary elliptic curves.

Findings

Constructed several families of optimal locally repairable codes with length O(q+2√q).

Developed a general framework for codes with two recovering sets.

Achieved codes with length O(q^2+2q) and specific Singleton defect bounds.

Abstract

Locally repairable codes with availability have become essential components in modern large-scale distributed cloud storage systems and numerous other applications. In this paper, we focus on the construction of locally repairable codes with one or two recovering sets via elliptic function fields. Prior pioneering work by Li et al. (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) employed maximal supersingular elliptic curves to obtain several optimal (classical) locally repairable codes. In contrast, we consider ordinary elliptic curves with many rational points. This approach yields several new families of \(q\)-ary optimal locally repairable codes with length \(O(q+2\sqrt{q})\) and flexible locality. Consequently, our work broadens the selection of curves available for the construction of optimal locally repairable codes. Furthermore, we present a general framework for constructing locally repairable codes with two recovering sets via automorphism groups of elliptic function fields. To realize this framework, we devise a novel construction for determining the functions \(e_i\) in the construction of locally repairable codes. By employing both supersingular and ordinary elliptic curves, we obtain several families of locally repairable codes with two recovering sets. In particular, we construct a family of \(q^2\)-ary locally repairable codes with two recovering sets, achieving length \(O(q^2+2q)\) and Singleton-defect \(O\!\left(\frac{2\ell}{q^2+2q-8\ell}\right)\), where \(\ell \mid\mid q + 2\) with \(4\ell < q\).