On $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes and their weight enumerators
J. Korean Math. Soc. 2021 Vol. 58, No. 3, 571-595
Published online April 5, 2021
Printed May 1, 2021
Maheshanand Bhaintwal, Soumak Biswas
Indian Institute of Technology Roorkee; 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 $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-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. Some optimal $p$-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.
Keywords : $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-linear codes, $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes, Gray map, weight enumerators
MSC numbers : Primary 94B05, 94B15, 94B60
Supported by : The second author would like to thank Ministry of Human Resource Development (MHRD), India, for providing financial support.
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