Abstract: Functional magnetic resonance imaging (fMRI) experiments help to render correct statistical inferences on brain function. However, few theoretical constructions of efficient designs for these crucial experiments exist in the literature. Recent works on the construction of circulant orthogonal arrays using algebraic difference sets show promise. However, such arrays are limited by the assumption that any interactions between the effects of a hemodynamic response function (HRF) and its residual effects are negligible in magnitude estimations of BOLD signals collected from an fMRI. Therefore, we propose a theoretical construction of a circulant orthogonal array of high strength using an extension of the complete difference system. FMRI experiments conducted using our proposed design show that the main effects of the individual HRF on the signal are not biased by the main effects of other HRFs, or by the interaction with its residual effects. Several properties of this new class of designs are studied, and the statistical regression model associated with this class of designs is presented.

Key words and phrases: Circulant almost orthogonal arrays, complete difference system, design efficiency, hemodynamic response function.