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 the absolute state complexity of algebraic geometric codes

A trellis of a code is a labeled directed graph whose paths from the initial to the terminal state correspond to the codewords. The main interest in trellises is due to their applications in the decoding of convolutional and block codes. The absolute state complexity of a linear code C is defined in terms of the number of vertices in the minimal trellises of all codes in the permutation equivalence class of C. In this thesis, we investigate the absolute state complexity of algebraic geometric codes. We illustrate lower bounds which, together with the well-known Wolf upper bound, give a good idea about the possible values of the absolute state complexities of algebraic geometric codes. A key role in the analysis is played by the gonality sequence of the function field that is used in code construction

A survey of the state-of-the-art and focused research in range systems

In this one-year renewal of NASA Contract No. 2-304, basic research, development, and implementation in the areas of modern estimation algorithms and digital communication systems have been performed. In the first area, basic study on the conversion of general classes of practical signal processing algorithms into systolic array algorithms is considered, producing four publications. Also studied were the finite word length effects and convergence rates of lattice algorithms, producing two publications. In the second area of study, the use of efficient importance sampling simulation technique for the evaluation of digital communication system performances were studied, producing two publications

Generalized Bezout's Theorem and its applications in coding theory

This paper presents a generalized Bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more curves for a large class of plane curves. A new approach to determine a lower bound on the minimum distance (and also the generalized Hamming weights) for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bezout theorem. Examples of more efficient linear codes are constructed using the generalized Bezout theorem and the new approach. For d = 4, the linear codes constructed by the new construction are better than or equal to the known linear codes. For d greater than 5, these new codes are better than the known codes. The Klein code over GF(2(sup 3)) is also constructed

An Introduction to Algebraic Geometry codes

We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes