On cyclic codes over Zq[u]/(u2) and their enumeration

In this study, we determine the structure of cyclic codes over the ring Zq[u]/(u2) which is isomorphic to R = Zq + uZq where q = ps, p is a prime, s is a positive integer, and u2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R[x]/(xn - 1), which is addressed in this paper completely. By establishing the structure of ideals of R[x]/(xn - 1) with gcd(p, n) = 1, we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size.(c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

In this study, the structure of cyclic codes over the ring Zq[u]/(u2) which is isomorphic to R = Zq + uZq is determined. The algebraic structure of ideals of the polynomial quotient ring R[x]/(xn - 1) is completely addressed, and an exact formula that enumerates the number of ideals of this ring is presented. Furthermore, the size of some special families of cyclic codes for specific q is determined.