The root diagram for one-point AG codes arising from certain curves with separated variables

Heegard, Little and Saints introduced in [8] an encoding algorithm for a class of AG codes via Gröbner basis more compact compared with the usual encoding via generator matrix. So, knowing that the main drawback of Gröbner basis is the high computational cost required for its calculation, in [12], the same authors introduced the concept of root diagram that allows the construction of an algorithm for computing a Gröbner basis with a lower complexity for one-point Hermitian codes. In [4], Farrán, Munuera, Tizziotti and Torres extended the results obtained in [12] for codes on norm-trace curves. In this work we generalize these results by constructing the root diagram for codes arising from certain curves with separated variables that has certain special automorphism and a Weierstrass semigroup generated by two elements. Such family of curves includes the norm-trace curve, among other curves with recent applications in coding theory

Higher Hamming weights for locally recoverable codes on algebraic curves

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds of distance proposed in [1] for some Hermitian LRC codes. [1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic curves. arXiv preprint arXiv:1501.04904, 2015

A new method for constructing small-bias spaces from Hermitian codes

We propose a new method for constructing small-bias spaces through a combination of Hermitian codes. For a class of parameters our multisets are much faster to construct than what can be achieved by use of the traditional algebraic geometric code construction. So, if speed is important, our construction is competitive with all other known constructions in that region. And if speed is not a matter of interest the small-bias spaces of the present paper still perform better than the ones related to norm-trace codes reported in [12]