Coding theory via Groebner bases

Abstract

Coding theory plays an important role in efficient transmission of data over noisy communication channels. It consists of two steps; the first step is to encode the data to reduce its sensitivity to noise during transmission, and the second step is to decode the received data by detecting and correcting the noise induced errors. In this thesis an algebraic approach is used to develop efficient encoding and decoding algorithms for a very commonly used class of linear codes, the Reed-Muller codes and the Golay codes. To develop the approach first the algebraic structure of linear codes is explored. For this, the reduced Groebner basis for a class of ideals in commutative polynomial rings is constructed. The extension of these ideals to a residue class ring enabled us to find the parameters of the corresponding codes. It is found that the corresponding codes contains the primitive Reed-Muller codes. The added advantage of this approach is that, once these Groebner bases are constructed a standard procedure can be used to develop encoding and decoding processes. A binomial ideal, defined as a sum of toric ideal and a prime ideal over some arbitrary field, is explored. It is shown that this ideal is equal to a binomial ideal over a prime field. Purpose of proving this equivalence is to study binary codes associated to this ideal. Minimal generators and Groebner basis found for this ideal showed that the situation is quite closely related to the toric case. The investigation of universal Groebner basis, Graver basis and circuits for this ideal revealed that they have the same relationship among them which is true in general for toric ideals. Each linear code can be described as a binomial ideal defined above. Since the reduced Groebner basis for any ideal plays a vital role in describing encoding and decoding processes for the corresponding codes, a natural reduced Groebner basis for this ideal is proposed for any general term order. In fact, if a generator matrix is given for any code, by constructing the corresponding particular binomial ideal, one can immediately describe the reduced Groebner basis. Information positions and parity check positions are then given by standard and non-standard monomials for the ideal. A systematic encoding algorithm for such codes is explained in terms of remainders of the information word computed with respect to the reduced Groebner basis. Furthermore, the binary and ternary Golay codes are studied algebraically in terms of the binomial ideal. Finally, a presentation of the binomial ideal of a linear code in terms of its syzygy modules is provided and the corresponding finite free resolution has been described.