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

Abstract

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

“…We find that the rate of the fiber product codes is extremely close to the product code construction bound, matching up to at least 4 decimal places in each case. The lower bound on minimum distances of these codes are also larger than that of the product code and Wang et al construction when (p, t) is not (3,2). This larger minimum distance comes at a cost of much greater length, however; the relative minimum distance of the fiber product codes is less than either of the other constructions.…”

“…In [5], the authors generalize the fiber product construction to t ≥ 2 and refine the parameters of the resulting codes. 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

“…, 3; e 2 = 0, 1, 2}, evaluated at points in B, is an [240, 60, 62]-locally recoverable code with availability 2. Every coordinate in a codeword can be recovered using two possible sets: one with 3 elements and another with 4 elements, giving a locality of (3,4).…”

“…In [5], the authors generalize the fiber product construction to t ≥ 2 and refine the parameters of the resulting codes. 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.…”

Isometry-Dual Flags of Many-Point AG Codes

“…In these codes, the relation between a code and its dual is given in terms of equivalence of certain divisors. Many recent explicit constructions and applications of these codes can be found in [1,22] and the references therein. Let F /F q be a function field.…”