Research On Error-correcting Codes Over Finite Rings In Information Safety

Error-correcting coding theory is the theoretieal bases of inofmration safety. At present,error-correcting coding theory over finite fields has been not only developed perfecting but also applied widely to the productive practice. With the successive development of production technique and the successive deepgoing researches on theory. Researches on error-correcting coding theory over finite rings have not only important theoretical significance but also important practical value. This dissertation makes a deepgoing research of various properties of linear codes, cyclic codes and one-weight codes as well as two-weight codes. The details are given as follws.1. We firstly give the structure of generator matrixes of the linear codes over ring F2 + vF2 , v 2=v. Secondly, we not only prove that the Gray images of the linear codes over ring F2 + vF2 are also linear codes, but also prove that the Gray images of the dual codes are also dual. Thirdly, we present various formulas of the codes weight between codes and their Gray images, and then give their Macwilliams identities. We also give generator polynomials of cyclic codes and their dual over ring Fp + vFp ,v 2= v, and determine their idempotent generators over ring F2 + vF2 , v 2=v. We prove that they are pricipal ideal generated. We also study Gray images of cyclic codes over ring F2 + vF2as well.2. We discuss the structures and propostitions of linear codes and their dual over ring Fp + vFp , v2= 1. By using The Chinese Remainder Theorem, we study the generator polynomials of cyclic codes and their dual over ring Fp + vFp. We prove that they are pricipal ideal generated. Moreover their Gray images are p-ary quasicyclic codes. Some optimal ternary quasi-cyclic codes are also constructed. Several counting formulas of weight distributions of linear codes over ring Fp + vFp are defined. By the relationship of linear codes and their dual codes over F p and the proposition of the Gray map, the MacWilliams identities between the linear codes and their dual codes are given.3. We study the structures of cyclic codes and constacyclic codes of odd length n over ring Fq + uFq + + u s?1Fq and determine the ranks of cyclic codes and their minimal generating sets.4. The Homogeneous weight over ring R = F2 + uF2 + +u k?1F2 is defined. Hamming distances and Homogeneous distances of (1 + u)? constacyclic codes of length 2s over the ring R are studied. By means of the theory of finite rings, the structure of (1 + u)?constacyclic codes of length 2s over R is also obtained. Especially, the structure and the size of cyclic self-dual codes over the ring are also given. Then, using the structure of such constacyclic codes, the distributions of the Hamming distances and Homogeneous distances of such constacyclic codes are determined. By serious analysis for the structure of cyclic codes of length 2e over ring F2 + uF2, we determine the Hamming distances and Lee distances of cyclic codes of length 2e over ring F2 + uF2.5. We discuss the structures and proposititions of one-Lee weight codes ring Z 4, and obtain some good binary one-Hamming weight codes under the Gray images. Moreover, using different method, we study the structures and proposititions of one-Homogeneous codes ring Z pm. Especially we consider the Z 8 and Z 9 case, and we obtain some best-known binary and ternary one-Hamming codes.