The Application Of Gr(?)bner Basis Theory To The Decoding Of Two Kinds Of Codes

In this paper, we summarize and study the application of the theory of Grobner basis to the decoding of cyclic codes over the Galois field Fq and of negacyclic codes over Z4.(1) Let C be a cyclic code over a Galois field Fq satisfying the minimum Ham-ming distance dH{G)≥2t+1and minimum Hamming weight ωH(e)<t. We summarize and research the decoding procedure using the Grobner basis theory on elimination ideals to determine the number of errors in a received word and summa-rize the algorithm to construct a new Grobner basis with respect to the term order <1from that of another term order<2.A decoding approach is naturally presented thereafter. Finally, the author gives a detailed evaluation for the decoding procedure in chapter3from different perspectives.(2) We summarize and research the procedure to find a Grobner basis which contains a minimal regular element of the solution module M={(a,b)|(1+T)a=b mod zt+1} applying the method of solution by approximations. Then, the author gives a key theorem to determine error locations and error values, and a decoding algorithm is therefore listed.