New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences

We present new constructions for perfect and odd perfect sequences over the quaternion group Q(8). In particular, we show for the first time that perfect and odd perfect quaternion sequences exist in all lengths 2(t) for t >= 0. In doing so we disprove the quaternionic form of Mow's conjecture that the longest perfect Q(8)-sequence that can be constructed from an orthogonal array construction is of length 64. Furthermore, we use a connection to combinatorial design theory to prove the existence of a new infinite class of Williamson sequences, showing that Williamson sequences of length 2(t) n exist for all t >= 0 when Williamson sequences of odd length n exist. Our constructions explain the abundance of Williamson sequences in lengths that are multiples of a large power of two.