| The article "A mathematical theory of communication" by Shannon, made the beginning of coding theory, and also proved the existence of good codes. From then on, people began to study the design of good codes. In the1990s, since some efficient binary nonlinear code was discovered, the coding theory over finite rings have been given more attention. Simplex codes is one of the hot topics of the study of error-correcting codes. Simplex codes with strict algebraic structure, so they have many good properties. Especially, the simplex codes can get a lot of good codes. In this paper, we mainly study simplex codes over several finite rings, details are given as follows:1. It introduces the generator matrices, the array consisting of all codewords, the structure of codewords, the torsion code, the weight distributions, the weight enumerator,the Gray map and Gray images of the simplex codes over the ring Z4.2. It redefines the generalized Lee weights, and defines the generalized Gray map of literature [16] over ring It also determines the generator matrices, the array consisting of all codewords, the structure of codewords, the torsion code, the weight distributions, the weight enumerator,the Gray map and Gray images of the simplex codes of types α and β over the ring3. It defines the Chinese Euclidean weights over the ring Z6, also introduces the generator matrices, the array consisting of all codewords, the structure of codewords, the torsion code, the weight distributions, the weight hierarchy and the weight enumerator of the simplex codes of types α,β and γ over the ring Z6.4. It defines the Lee weights, the Euclidean weights and the Bachoc weights, also determines the generator matrices, the array consisting of all codewords, the structure of codewords, the torsion code and the weight distributions of the simplex codes of types α and determine the generator matrices, the structure of codewords and the weight distributions of the simplex codes of types γ over the ring F4+vF4. |
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