A new type of $m$-class cyclic association scheme is defined for $v$ objects, where $v$ is either a prime or power of a prime. The two cases, when $v$ is even and when $v$ is odd are dealt with separately. A simple method of construction of PBIB designs based on this scheme is proposed. The PBIB designs presented in this paper have cyclic solutions and are obtainable through more than one initial block. The proposed scheme has been called an $NC_m$ association scheme and the number of independent $p_{jk}$'s for this scheme is obtained.