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arXiv · 1905.00267

New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences

Curtis Bright·Ilias Kotsireas·Vijay Ganesh
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Abstract

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 $t\geq0$. 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 $t\geq0$ 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.

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Curtis Bright, Ilias Kotsireas, Vijay Ganesh. 2019-05-01. New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences. https://doi.org/10.1109/tit.2020.3016510

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