### 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 |
---|---|

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

### Cite this

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_{n}× Q

_{8}'

*Cryptography and Communications: discrete structures, Boolean functions and sequences*, vol. 10, no. 2, pp. 357-368. https://doi.org/10.1007/s12095-017-0224-y

**Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in C _{n} × Q _{8}.** / Barrera Acevedo, Santiago; Dietrich, Heiko.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q 8

AU - Barrera Acevedo, Santiago

AU - Dietrich, Heiko

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±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 Q8 ∪ qQ8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2; here Cm is a cyclic group of order m. We show that if n = pa + 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 Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 ∪ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 ∪ qQ8).

AB - Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±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 Q8 ∪ qQ8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2; here Cm is a cyclic group of order m. We show that if n = pa + 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 Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 ∪ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 ∪ qQ8).

KW - Hadamard matrices

KW - Perfect sequences

KW - Quaternions

KW - Relative difference sets

UR - http://www.scopus.com/inward/record.url?scp=85042130110&partnerID=8YFLogxK

U2 - 10.1007/s12095-017-0224-y

DO - 10.1007/s12095-017-0224-y

M3 - Article

VL - 10

SP - 357

EP - 368

JO - Cryptography and Communications: discrete structures, Boolean functions and sequences

JF - Cryptography and Communications: discrete structures, Boolean functions and sequences

SN - 1936-2447

IS - 2

ER -