D-optimal cyclic complex linear designs and supplementary difference sets with association vector

Abstract

A linear model with one treatment at V levels and the first-order regression on k continuous covariates with values on the k-cube is considered. We restrict our attention to the subclass of designs having k = V and equal number r of observations per treatment level and for which the allocation matrix of each treatment level is obtained through a cyclic permutation of the rows of the allocation matrix of the first treatment level. These designs are called ‘equireplicated cyclic’ complex linear designs. Using cyclic Hadamard matrices, Golay sequences and Williamson type matrices, we construct D-optimal equireplicated cyclic designs, attaining the ‘Hadamard upper bound’ (H.u.b.). Supplementary difference sets r − {V; k 1, k 2, …, k r; λ; (i r 2 ) V 1} with association vector (i r 2 ) V 1 are introduced to construct D-optimal equireplicated cyclic designs for N ≡ r V ≡ 0 mod 4 observations, r ≡ 0 mod 4, where λ = rV 4 and k j 's are defined through V, j = 1,2, …, r. An infinite series of such r-SDSAV is given when V = 1 + q + q 2 is a prime and q is a prime power. Also, an exhaustive search algorithm is developed for constructing such nonequivalent supplementary difference sets 4 − { V; k 1, k 2, k 3, k 4; λ; ( i 2) V 1}. The algorithm is applied for 4 ⩽ V < 16 and for all decompositions of 4 V into four squares having zero sum.