New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences

05/01/2019

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by   Curtis Bright, et al.

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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.