Locally recoverable codes from algebraic curves with separated variables

Abstract

A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.2010 Mathematics Subject Classification. 94B27, 11G20, 11T71, 14G50, 94B05.

Locally Recoverable Codes From Automorphism Group of Function Fields of Genusg≥ 1

“…Many constructions of LRC codes from function fields arose in recent years, over the rational function field [14], [15], elliptic function fields [19] and over algebraic curves (see [22], [28]), fiber product of curves [11] and curves with separated variables [23]. In a variant of the different ways to construct LRC codes, we focus on codes from algebraic function fields of genus g ≥ 1 using certain subgroups of the automorphism group of the underlying curve.…”

“…In recent years, the study of locally reparable codes has attracted a lot of attention. Most of the results concern bounds on the minimum distance [9], [18], [27], [28] and construction of LRC codes [5], [11], [14]- [16], [19], [20], [22], [23], [26], [28].…”

Minimum distance and parameter ranges of locally recoverable codes with availability from fiber products of curves

“…The group-theoretic method of constructing locally recoverable codes with many recovery sets has also been studied, notably in [3]. The general approach of creating locally recoverable codes from rational maps is pursued in [7] and extended to algebraic curves defined by equations with separated variables in [8], but the general fiber product construction still requires more exploration.…”

Minimum Distance and Parameter Ranges of Locally Recoverable Codes with Availability from Fiber Products of Curves

“…The group-theoretic method of constructing locally recoverable codes with many recovery sets has also been studied, notably in [3]. The general approach of creating locally recoverable codes from rational maps is pursued in [7] and extended to algebraic curves defined by equations with separated variables in [8], but the general fiber product construction still requires more exploration.…”