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 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 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.