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

The second Feng-Rao number for codes coming from telescopic semigroups

In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Ap\'ery sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound

An Introduction to Algebraic Geometry codes

We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes

Relative generalized Hamming weights of one-point algebraic geometric codes

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights

Johnson type bounds for mixed dimension subspace codes

Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.Comment: 16 pages, typos correcte