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 $t\geq 0$. 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 $t\geq 0$ 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. [ABSTRACT FROM AUTHOR]