118 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
The Weight Hierarchies of Linear Codes from Simplicial Complexes
The study of the generalized Hamming weight of linear codes is a significant
research topic in coding theory as it conveys the structural information of the
codes and determines their performance in various applications. However,
determining the generalized Hamming weights of linear codes, especially the
weight hierarchy, is generally challenging. In this paper, we investigate the
generalized Hamming weights of a class of linear code \C over \bF_q, which
is constructed from defining sets. These defining sets are either special
simplicial complexes or their complements in \bF_q^m. We determine the
complete weight hierarchies of these codes by analyzing the maximum or minimum
intersection of certain simplicial complexes and all -dimensional subspaces
of \bF_q^m, where 1\leq r\leq {\rm dim}_{\bF_q}(\C)
The weight hierarchies and chain condition of a class of codes from varieties over finite fields
The generalized Hamming weights of linear codes were first introduced by Wei. These are fundamental parameters related to the minimal overlap structures of the subcodes and very useful in several fields. It was found that the chain condition of a linear code is convenient in studying the generalized Hamming weights of the product codes. In this paper we consider a class of codes defined over some varieties in projective spaces over finite fields, whose generalized Hamming weights can be determined by studying the orbits of subspaces of the projective spaces under the actions of classical groups over finite fields, i.e., the symplectic groups, the unitary groups and orthogonal groups. We give the weight hierarchies and generalized weight spectra of the codes from Hermitian varieties and prove that the codes satisfy the chain condition
Generalized weights and bounds for error probability over erasure channels
New upper and lower bounds for the error probability over an erasure channel
are provided, making use of Wei's generalized weights, hierarchy and spectra.
In many situations the upper and lower bounds coincide and this allows
improvement of existing bounds. Results concerning MDS and AMDS codes are
deduced from those bounds
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