Thesis (MSc (Mathematics))--University of Stellenbosch, 2006. / The main topic of this thesis is the construction of the algebraic geometric codes (Goppa codes), and their decoding by the list-decoding, which allows one to correct beyond half of the minimum distance. We also consider the list-decoding of the Reed–Solomon codes as they are subclass of the Goppa codes, and the determination of the parameters of the non primitive BCH codes. AMS Subject Classification: 4B05, 94B15, 94B35, 94B27, 11T71, 94B65,B70. Keywords: Linear codes, cyclic codes, BCH codes, Reed–Solomon codes, list-decoding, Algebraic Geometric codes, decoding, bound on codes, error probability.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-11-06T18:45:09Z No. of bitstreams: 1 maurorodriguesrochajunior.pdf: 550118 bytes, checksum: 5b26ad1ab2bd9d4a190d742762346968 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-11-09T14:32:38Z (GMT) No. of bitstreams: 1 maurorodriguesrochajunior.pdf: 550118 bytes, checksum: 5b26ad1ab2bd9d4a190d742762346968 (MD5) / Made available in DSpace on 2017-11-09T14:32:38Z (GMT). No. of bitstreams: 1 maurorodriguesrochajunior.pdf: 550118 bytes, checksum: 5b26ad1ab2bd9d4a190d742762346968 (MD5) Previous issue date: 2017-08-11 / O principal objetivo desse trabalho é estudar duas aplicações distintas das bases de Gröbner a códigos lineares. Com esse objetivo, estudamos como relacionar códigos a outras estruturas matemáticas, fazendo com que tenhamos novas ferramentas para a realização da codificação. Em especial, estudamos códigos cartesianos afins e os códigos algébrico-geométricos de Goppa. / The main objective of this work is to study two different applications of Gröbner basis to linear codes. With this purpose, we study how to relate codes to other mathematical structures, allowing us to use new tools to do the coding. In particular, we study affine cartesian codes e algebraic-geometric Goppa codes.