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

Abstract

We construct families of locally recoverable codes with availability t ≥ 2 using fiber products of curves, determine the exact minimum distance of many families, and prove a general theorem for minimum distance of such codes. The paper concludes with an exploration of parameters of codes from these families and the fiber product construction more generally. We show that fiber product codes can achieve arbitrarily large rate and arbitrarily small relative defect, and compare to known bounds and important constructions from the literature.

On cyclic algebraic-geometry codes

“…Research in codes in recent years has proven to be a highly productive field of investigation. For instance, construction of locally recoverable codes over maximal curves can be found in [3] or [7], AG-codes over the maximal Beelen-Montanucci curve [20], weights of dual codes over the maximal GK-curve [2], lifting of AG-codes in [8]. The maximal Hermitian curve was used for applications to quantum codes [9], self orthogonal maximum distance separable codes [21], investigation of the isometry-dual property in flags of codes [6], construction of codes using places of higher degree [22], and many point codes in [18].…”