# New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences

## TL;DR

This paper introduces new infinite families of perfect quaternion sequences of lengths 2^t, disproves a conjecture about their maximum length, and establishes the existence of a broad class of Williamson sequences linked to combinatorial design theory.

## Contribution

It provides the first constructions of perfect quaternion sequences for all lengths 2^t and links Williamson sequences to combinatorial design theory, expanding known sequence families.

## Key findings

- Perfect quaternion sequences exist for all lengths 2^t.
- Disproves Mow's conjecture on maximum sequence length.
- Establishes infinite classes of Williamson sequences for lengths multiple of powers of two.

## 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.