On $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes and their weight enumerators

J. Korean Math. Soc. Published online April 5, 2021

Maheshanand Bhaintwal and Soumak Biswas
Indian Institute of Technology, Roorkee

Abstract : In this paper we study the algebraic structure of $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes, where $u^k=0$ and $p$ is a prime. A $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-linear code of length $(r+s)$ is an $R_k$-submodule of $\mathbb{Z}_p^r \times R_k^s$ with respect to a suitable scalar multiplication, where $R_k = \mathbb{Z}_p[u]/\langle u^k\rangle$. Such a code can also be viewed as an $R_k$-submodule of $\mathbb{Z}_p[x]/\langle x^r-1\rangle \times R_k[x]/\langle x^s-1\rangle$. A new Gray map has been defined on $\mathbb{Z}_p[u]/\langle u^k\rangle$. We have considered two cases for studying the algebraic structure of $\mathbb{Z}_p\mathbb{Z}_p[u^k]$-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered $(r,p)=1$ and $(s,p)\neq 1$, and in the second case we consider $(r,p)=1$ and $(s,p)=1$. We have established the MacWilliams identity for complete weight enumerators of $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-linear codes. Examples have been given to construct $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes, through which we get codes over $\mathbb{Z}_p$ using the Gray map. An example has also been given to illustrate the use of MacWilliams identity.