15 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
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 and extended weight polynomials of almost affine codes
This is a post-peer-review, pre-copyedit version of an article published in Lecture Notes in Computer Science, International Castle Meeting on Coding Theory and Applications ICMCTA 2017: Coding Theory and Applications, 207-216. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-319-66278-7_17 .This paper is devoted to giving a generalization from linear
codes to the larger class of almost affine codes of two different results.
One such result is how one can express the relative generalized Hamming
weights of a pair of codes in terms of intersection properties between the
smallest of these codes and subcodes of the largest code. The other result
tells how one can find the extended weight polynomials, expressing the
number of codewords of each possible weight, for each code in an infinite
hierarchy of extensions of a code over a given alphabet. Our tools will
be demi-matroids and matroids
Message Randomization and Strong Security in Quantum Stabilizer-Based Secret Sharing for Classical Secrets
We improve the flexibility in designing access structures of quantum
stabilizer-based secret sharing schemes for classical secrets, by introducing
message randomization in their encoding procedures. We generalize the
Gilbert-Varshamov bound for deterministic encoding to randomized encoding of
classical secrets. We also provide an explicit example of a ramp secret sharing
scheme with which multiple symbols in its classical secret are revealed to an
intermediate set, and justify the necessity of incorporating strong security
criterion of conventional secret sharing. Finally, we propose an explicit
construction of strongly secure ramp secret sharing scheme by quantum
stabilizers, which can support twice as large classical secrets as the
McEliece-Sarwate strongly secure ramp secret sharing scheme of the same share
size and the access structure.Comment: Publisher's Open Access PDF. arXiv admin note: text overlap with
arXiv:1811.0521
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)