Proof of Williamsʹ Conjecture on Experimental Designs Balanced for Pairs of Interacting Residual Effects

E. J. Williams conjectured that for every prime power v ⩾ 3 there exists a v(v - 1) × v design balanced for pairs of interacting residual effects, and he proved this conjecture for the case in which v is a prime. Later, this conjecture was verified for v ⩽ 64 by machine calculations. We prove the conjecture of Williams by a construction based on maximal period sequences in the finite field GF(v) and on certain complete sets of orthogonal latin squares.