1,085 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
Erasure List-Decodable Codes from Random and Algebraic Geometry Codes
Erasure list decoding was introduced to correct a larger number of erasures
with output of a list of possible candidates. In the present paper, we consider
both random linear codes and algebraic geometry codes for list decoding erasure
errors. The contributions of this paper are two-fold. Firstly, we show that,
for arbitrary ( and are independent),
with high probability a random linear code is an erasure list decodable code
with constant list size that can correct a fraction
of erasures, i.e., a random linear code achieves the
information-theoretic optimal trade-off between information rate and fraction
of erasure errors. Secondly, we show that algebraic geometry codes are good
erasure list-decodable codes. Precisely speaking, for any and
, a -ary algebraic geometry code of rate from the
Garcia-Stichtenoth tower can correct
fraction of erasure errors with
list size . This improves the Johnson bound applied to algebraic
geometry codes. Furthermore, list decoding of these algebraic geometry codes
can be implemented in polynomial time
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
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