Abstract
In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 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 Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.
Highlights
Error correcting codes and error detection codes play an important role in data networks and satellite applications
The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is obtained and we prove that the Gray map of the linear code over the ring Z15 is linear
We prove that the minimal Lee weight of C is equal to the minimal Hamming weight of its Gray Mapping
Summary
Error correcting codes and error detection codes play an important role in data networks and satellite applications. Since the 1970s, there are many research papers about codes over the finite ring. The importance of finite rings in algebraic coding theory was established in the early 1990s by observing that some non-linear binary codes allow a linear representation of Z4 (see [1] [7]). The homogeneous weight can be a natural extension of the Hamming weight of the code over finite rings. We will concern the linear code over the ring Z15, which has p ⋅ q elements and p ≠ q. We get the ideals of the ring Z15 through its non-units and give the Lee weights of elements in Z15. The linear property of the Gray mapping of a linear code is obtained
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