ON ℤ p ℤ p [u]/ k >-CYCLIC CODES AND THEIR WEIGHT ENUMERATORS

In this paper we study the algebraic structure of ZpZp[u]/ 〈uk〉-cyclic codes, where uk = 0 and p is a prime. A ZpZp[u]/〈u〉-linear code of length (r + s) is an Rk-submodule of Zp × Rs k with respect to a suitable scalar multiplication, where Rk = Zp[u]/〈u〉. Such a code can also be viewed as an Rk-submodule of Zp[x]/〈x−1〉×Rk[x]/〈x−1〉. A new Gray map has been defined on Zp[u]/〈u〉. We have considered two cases for studying the algebraic structure of ZpZp[u]/〈u〉-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) 6= 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 ZpZp[u]/〈u〉-linear codes. Examples have been given to construct ZpZp[u]/〈u〉-cyclic codes, through which we get codes over Zp 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.