491 research outputs found
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
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
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
Minimum-weight codewords of the Hermitian codes are supported on complete intersections
Let be the Hermitian curve defined over a finite field
. In this paper we complete the geometrical characterization
of the supports of the minimum-weight codewords of the algebraic-geometry codes
over , started in [1]: if is the distance of the code, the
supports are all the sets of distinct -points on
complete intersection of two curves defined by polynomials with
prescribed initial monomials w.r.t. \texttt{DegRevLex}.
For most Hermitian codes, and especially for all those with distance studied in [1], one of the two curves is always the Hermitian curve
itself, while if the supports are complete intersection of
two curves none of which can be .
Finally, for some special codes among those with intermediate distance
between and , both possibilities occur.
We provide simple and explicit numerical criteria that allow to decide for
each code what kind of supports its minimum-weight codewords have and to obtain
a parametric description of the family (or the two families) of the supports.
[1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections,
arXiv preprint arXiv:1510.03670 (2015)
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