Generalized orthogonal designs have many useful applications in statistics, web engineering, and coding theory along with signal processing. In this paper, for every prime number  we develop a method of constructing a set of four circulant amicable matrices  satisfying   is the identity matrix of order n) over the prime field  Using these matrices with T-matrices, we suggest some approaches to construct infinite classes of generalized orthogonal designs (GODs) over the prime fields

orthogonal designs, generalized orthogonal designs, amicable matrices, T-matrices, prime field.