Author(s)

Wensheng Li^*, Lingyu Wan, Mengtian Yue, Wei Chen, Xuedong Zhang

Langfang Normal University, Langfang, China.

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.

Lee Weight, Hamming Weight, Gray Map, Dual Code

Li, W. , Wan, L. , Yue, M. , Chen, W. and Zhang, X. (2020) Linear Codes over the Finite Ring Z₁₅. Advances in Linear Algebra & Matrix Theory, 10, 1-5. doi: 10.4236/alamt.2020.101001.