On Reed-Solomon and algebraic geometry codes

Reed-Solomon codes are an important class of codes both in theory and in practice. The central theme of the thesis is an investigation of Reed-Solomon codes, and their decoding algorithms, modifications of Reed-Solomon codes and an interpretation of certain algebraic geometry codes in terms of Reed-Solomon codes. It is shown that, in a straightforward manner, new codes with a given length and dimension that have minimum distance one less than maximum distance separable codes, can be constructed from Reed-Solomon codes.;A class of algebraic geometry codes known as Hermitian codes is shown to be special combinations of concatenated generalized Reed-Solomon codes. This interpretation yields insight into the structure of Hermitian codes. A decoding algorithm for codes which have a large easily decodable subcode is given. Encoding and decoding algorithms are presented for Hermitian codes.;A new class of codes from hyperelliptic curves is introduced. Derivation of the parameters of these codes does not use results from algebraic geometry. A new method of obtaining minimum weight vectors is presented. A decoding algorithm for these codes is derived which can be understood in terms of Reed-Solomon decoding.;Two new and efficient decoding algorithms are given for Reed-Solomon codes. The first algorithm uses a Lagrange iterpolation polynomial and applies the extended Euclidean algorithm to the decoding problem. This algorithm is referred to as the Lagrange-Euclidean decoding algorithm and has complexity comparable to existing decoding algorithms. The second decoding algorithm operates directly on the remainder of the received polynomial when divided by the generator polynomial of the code and is efficient for correcting a small number of errors.