There are Infinitely Many Williams Squares with Circular Structure and Order a Multiple of 4

Williams squares, also known as row quasi-complete Latin squares, that have circular structure are used in repeated measurements designs where error terms are correlated. They are known to exist for all orders that are not multiples of 4 and also order 8; there is no such square of order 4. We use a variation of the generating array method to construct a Williams square of order 12 with circular structure. We also give a product theorem that produces Williams squares of orders 8r and 12s, where the smallest prime divisor of r is at least 11 and the smallest prime divisor of s is at least 13, that have circular structure.