Locally recoverable codes with availability <i>t</i>≥2 from fiber products of curves

Abstract

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlȃduţ [4] to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

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].…”

“…In particular, we obtain families of locally repairable codes with good relative parameters, since the relative defect ∆(C) as in (4) tends to zero when q goes to infinity; see Propositions IV.2, IV.4, IV.6, IV.8, and IV.9. Also, we compare the parameters of some of these LRC codes with the ones constructed in [11] obtained from fiber products of algebraic curves; see Remark IV.7.…”

Codes With Hierarchical Locality From Covering Maps of Curves

“…. , t in (8). Let us construct a code C by evaluating these functions at the points in D as described in (9).…”

“…To generate H-LRC codes with availability we use a construction inspired by the LRC codes with availability introduced in [3] and developed in [8].…”