In this paper, the structure of the non-chain ring Z 15 is studied. The ideals of the ring Z 15 are obtained through its non-units and the Lee weights of elements in Z 15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z 15 . Further, the Gray mapping from 15 n  to 2 15 n  is defined and it's shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z 15 is also obtained and we prove that the Gray map of the linear code over the ring Z 15 is also linear. IntroductionError correcting codes and error detection codes play an important role in data networks and satellite applications. Most coding theory is interested. Linear code has a clear structure and it is easy to find, understand, edit and decode for codes over finite rings. Since the 1970s, there are many research papers about codes over the finite ring. Several good nonlinear binary codes have been discovered.The circulation code on Z 4 is composed of a Gary mapping structure [1]. After that, many researchers carried out more and more research on the code of finite ring [2] [3] [4] [5] [6]. The importance of finite rings in algebraic coding theory was established in the early 1990s by observing that some non-linear binary codes actually allow a linear representation of Z 4 (see [1] [7]). It is also noted that the codes on the ring are particularly useful, if the distance function in the alphabet is not given by the usual Hamming metric, but by the homogeneous weight [8]. Examples of homogeneous weights are Hamming weights on finite fields and Lee weights on Z 4 . The homogeneous weight can be a natural extension of the Hamming weight of the code over finite rings.

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Linear codes are considered over the ring Z 8 +uZ 8 , a non chain extension of Z 8 . Gray maps, the complete enumerators for these codes are defined and MacWilliams identities for the complete enumerators are obtained.