907 research outputs found
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
Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth's second tower
Asymptotically good sequences of ramp secret sharing schemes were given in
[Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] by using
one-point algebraic geometric codes defined from asymptotically good towers of
function fields. Their security is given by the relative generalized Hamming
weights of the corresponding codes. In this paper we demonstrate how to obtain
refined information on the RGHWs when the codimension of the codes is small.
For general codimension, we give an improved estimate for the highest RGHW
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
On asymptotically good ramp secret sharing schemes
Asymptotically good sequences of linear ramp secret sharing schemes have been
intensively studied by Cramer et al. in terms of sequences of pairs of nested
algebraic geometric codes. In those works the focus is on full privacy and full
reconstruction. In this paper we analyze additional parameters describing the
asymptotic behavior of partial information leakage and possibly also partial
reconstruction giving a more complete picture of the access structure for
sequences of linear ramp secret sharing schemes. Our study involves a detailed
treatment of the (relative) generalized Hamming weights of the considered
codes
On the similarities between generalized rank and Hamming weights and their applications to network coding
Rank weights and generalized rank weights have been proven to characterize
error and erasure correction, and information leakage in linear network coding,
in the same way as Hamming weights and generalized Hamming weights describe
classical error and erasure correction, and information leakage in wire-tap
channels of type II and code-based secret sharing. Although many similarities
between both cases have been established and proven in the literature, many
other known results in the Hamming case, such as bounds or characterizations of
weight-preserving maps, have not been translated to the rank case yet, or in
some cases have been proven after developing a different machinery. The aim of
this paper is to further relate both weights and generalized weights, show that
the results and proofs in both cases are usually essentially the same, and see
the significance of these similarities in network coding. Some of the new
results in the rank case also have new consequences in the Hamming case
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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