Abstract A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K -cube is considered. We restrict our attention to classes of designs d for which the number of observations N to be taken is a multiple of V , i.e. N = V × R with R ≥2, and each treatment level is observed R times. Among these designs, called here equireplicated , there is a subclass characterized by the following: the allocation matrix of each treatment level (for short, allocation matrix ) is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. We call these designs cyclic . Besides having easy representation, the most efficient cyclic designs are often D-optimal in the class of equireplicated designs. A known upper bound for the determinant of the information matrix M ( d ) of a design, in the class of equireplicated ones, depends on the congruences of N and V modulo 4. For some combinations of parameter moduli, we give here methods of constructing families of D-optimal cyclic designs. Moreover, for some sets of parameters ( N, V,K = V ), where the upper bound on ∣ M ( d )∣ (for that specific combination of moduli) is not attainable, it is also possible to construct highly D-efficient cyclic designs. Finally, for N ≤24 and V ≤6, computer search was used to determine the most efficient design in the class of cyclic ones. They are presented, together with their respective efficiency in the class of equireplicated designs.