Locally Recoverable Codes with availability from a family of fibered surfaces

TL;DR

This paper introduces a new class of locally recoverable codes with availability 2, constructed from fibered surfaces, combining algebraic geometry and function field techniques, and provides a geometric interpretation for specific parameters.

Contribution

It presents the first error-correcting codes constructed from doubly elliptic (K3) surfaces, with explicit bounds and geometric insights for codes with locality 3.

Findings

Constructed LRCs with availability 2 from fibered surfaces.

Established a sharp minimum distance bound for locality 3.

Provided a geometric interpretation using doubly elliptic surfaces.

Abstract

We construct Locally Recoverable Codes (LRCs) with availability $2$ from a family of fibered surfaces. To obtain the locality and availability properties, and to estimate the minimum distance of the codes, we combine techniques coming from the theory of one-variable function fields and from the theory of fibrations on surfaces. When the locality parameter is $r=3$, we obtain a sharp bound on the minimum distance of the codes. In that case, we give a geometric interpretation of our codes in terms of doubly elliptic surfaces. In particular, this provides the first instance of an error correcting code constructed using a (doubly elliptic) K3 surface.