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We study locally recoverable codes on algebraic curves. In the first part of the manuscript, we provide a bound on the generalized Hamming weight of these codes. In the second part, we propose a new family of algebraic geometric LRC codes, which are LRC codes from the Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds on the distance proposed in Barg et al. (2015) [1] of some Hermitian LRC codes. [ABSTRACT FROM AUTHOR]

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any F q 2 . From these geometric characterizations, we obtain explicit formulas. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with d ≤ q and all second-weight codewords for distance- 3 , 4 codes. [ABSTRACT FROM AUTHOR]

Abstract: We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In particular, we provide explicit counting formulas that have also applications to some Hermitian codes. [Copyright &y& Elsevier]

In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field F q 2 . We focus on those with distance d ≥ q 2 − q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing μ q + λ (q + 1) of the distance d with μ , λ non negative integers, and μ ≤ q , and consider all the curves X of the affine plane A F q 2 2 of degree μ + λ defined by polynomials with x μ y λ as leading monomial with respect to the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A F q 2 2 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple F q 2 -points and there is a curve X such that Z coincides with the scheme theoretic intersection H ∩ X (namely, as a cycle, Z = H ⋅ X). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords. [ABSTRACT FROM AUTHOR]