### Abstract

Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q_{8} = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q_{8} ∪ qQ_{8} with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in C_{n} × Q_{8} with forbidden subgroup C_{2}; here C_{m} is a cyclic group of order m. We show that if n = p^{a} + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in C_{n} × Q_{8} with forbidden subgroup C_{2}. Lastly, we show that every perfect sequence of length n over Q_{8} ∪ qQ_{8} yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q_{8} ∪ qQ_{8}).

Original language | English |
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Pages (from-to) | 357-368 |

Number of pages | 12 |

Journal | Cryptography and Communications: discrete structures, Boolean functions and sequences |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

### Keywords

- Hadamard matrices
- Perfect sequences
- Quaternions
- Relative difference sets