[Show abstract][Hide abstract] ABSTRACT: There are two gradient descent decoding procedures for binary codes proposed independently by Liebler and by Ashikhmin and Barg. Liebler in his paper mentions that both algorithms have the same philosophy but in fact they are rather different. The purpose of this communication is to show that both algorithms can be seen as two ways of understanding the reduction process algebraic monoid structure related to the code. The main tool used for showing this is the Gröbner representation of the monoid associated to the linear code.
Information Theory Workshop (ITW), 2010 IEEE; 10/2010
[Show abstract][Hide abstract] ABSTRACT: A new algorithmic method is presented for the decoding process of a linear code. This method has the flavor of the FGLM techniques that belong to the framework of the Gröbner Bases Theory. This method consists of two algorithms, one for the prepa-ration phase of the code, and the other one for the decoding process. Starting from a parity check matrix of the linear code, our approach enables the user to decode a given codeword in a number of steps proportional to its length. Complexity analysis, complete examples and comments about the application of FGLM techniques to linear codes are also given in this paper.
[Show abstract][Hide abstract] ABSTRACT: We obtain a G-presentation of the symmetric group, by applying the Gröbner bases method. We compare the G-presentation obtained with the initial presentation (given by Carmichael) and two presentations (given by Le Chenadec).
[Show abstract][Hide abstract] ABSTRACT: We emphasize the main results and algorithms presented in [M. Borges-Quintana, F. Winkler and M. Borges-Trenard, FGLM techniques applied to linear codes: An algorithm for decoding linear codes, Techn. Report, RISC-Linz, RISC-00-14, J. Kepler Univ., Linz, Austria (2000); and An FGLM method for decoding linear codes, in Proc. Conf. EACA-2000, Barcelona, Spain, 117-128 (2000)] related with a new approach for decoding linear codes. This approach is derived from a mixture of the syndrome decoding method and the FGLM techniques. We also explain the meaning of “FGLM techniques”, and the possibilities for new applications are shown by our concrete situation concerning linear codes.
[Show abstract][Hide abstract] ABSTRACT: We present a methodology and algorithms for the obtainment of normal forms for the words of a finite presented monoid. This methodology bases on some result in the theory of formal languages.