Repeated-root Cyclic Codes Over Z_q And F_p+uF_p+…+u～(k-1)F_p

Cyclic codes are a widely studied family of codes . At present, with the successive development of production technique and the successive deep-going researches on cyclic coding theory over finite rings have not only important theoretical significance but also important practical value.This dissertation study Cyclic codes over finite rings of length N, where p divides N. This study has been shown an enormous interest in recent years. On ring Z_q (q = p^s), theauthor using the discrete Fourier transform, describe the structure and spectral representation of cyclic codes C of length N = p^kn over Z_q .Then the author compute theHamming weight of cyclic codes C. Ring F₂+uF₂ is a four-element ring which shares some good properties of both Z₄ and F₄. The ring F_p+uF_p+...+u^k-lF_p, which is the general caseof ring F₂+uF₂, has already been used in the construction of optimal frequency-hopping sequence by P. Udaya. Coding theory over these rings has recently received a great deal ofinterest among coding theorists. The author studies cyclic codes over R = forarbitrary length N. We shall use discrete Fourier transform to obtain an isomorphismγbetween and a direct sum .So, we give a method to represent an ideal in Galois ring S_u^k(m,ω), then obtaining the structure of cyclic codes over via this isomorphismγ. The inverse isomorphism ofγis explicitlydetermined, so that the polynomial representations of the corresponding ideals can be calculated.