Codis algebraics

Abstract

[en] Errors occur during the process of information transmission due to the channels through which the information travels. In this project we present methods through which to detect the given errors and correct them so as to guarantee the correct transmission of information. In this context, we will present the functioning of the lineal codes giving the codification tools as well as the decodification ones. The first decodification process we will work on is the one known as that of the ”syndrome”, and after that we will present the theory about lineal codes, focussing the Hamming codes and their benefits when it comes to the detection and correction of the commited errors. The second part of the project will focus on cyclic codes, a subset of lineal codes that bring more efficiency to the codification process. Among these we can find the Reed-Solomon codes, and at the same time we will give a new tool for their decodification. Moreover, we will observe that the cyclic codes idea can be extended to polynomial rings in multiple variables.