Research On Some Problems Of Cyclic Codes And Constacyclic Codes Over The Ring F_p^m+uF_p^m

Classical coding theory takes place in the setting of vector spaces over finite fields. In the 1900s, the theory of error-correcting codes over finite rings has experienced tremendous growth since the significant discovery that several well-known prominent families of good nonlinear binary codes can be identified as images of linear codes over Z 4 under the Gray map. Since then, codes over finite rings have been given more attention. In this paper, we study the structures and mass formulas of cyclic codes and constacyclic codes of length p k over the ring R = Fp m + uFpm, while the permutation-equivalences of constacyclic codes are also investigated. For the constacyclic codes with ( n, p ) = 1, their permutation–equivalences and structures of Gray images are discussed. The details are given as follows:1.Using equivalent permutation, we give a unique method of representing cyclic codes of length p k over the ring R . Based on that, the number of codewords of cyclic codes and the unique method of representing dual codes are provided;2. We discuss the equivalent conditions of cyclic self-dual codes of length p k, and give their mass formulas. Besides this, all cyclic self-dual codes are given when p = 3, k = 1 and k = 2;3. The permutation- equivalences of (1 +αu)-cyclic codes and (ξi +αu)-cyclic codes with ( n, p ) = 1 over the ring R are discussed and the structures of their Gray images are given.4. We study the permutation-equivalences and mass formulas ofα-cyclic codes and (uβ?α)-cyclic codes of length p k over the ring R , the mass formulas ofα-cyclic self-dual codes of length p k are also given.