### Abstract

Let G=Cn1×⋯×Cnm be an abelian group of order n= n_{1}⋯ n_{m}, where each Cnt is cyclic of order n_{t}. We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in G× Q_{8} relative to the centre Z(Q_{8}) and the perfect arrays of size n_{1}× ⋯ × n_{m} over the quaternionic alphabet Q_{8}∪ qQ_{8}, where q= (1 + i+ j+ k) / 2. In view of this connection, for m= 2 we introduce new families of relative difference sets in G× Q_{8}, as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.

Original language | English |
---|---|

Pages (from-to) | 397–406 |

Number of pages | 10 |

Journal | Mathematics in Computer Science |

Volume | 12 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2018 |

### Keywords

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

### Cite this

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**Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays.** / Barrera Acevedo, Santiago; Dietrich, Heiko.

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

TY - JOUR

T1 - Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

AU - Barrera Acevedo, Santiago

AU - Dietrich, Heiko

PY - 2018/12

Y1 - 2018/12

N2 - Let G=Cn1×⋯×Cnm be an abelian group of order n= n1⋯ nm, where each Cnt is cyclic of order nt. We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in G× Q8 relative to the centre Z(Q8) and the perfect arrays of size n1× ⋯ × nm over the quaternionic alphabet Q8∪ qQ8, where q= (1 + i+ j+ k) / 2. In view of this connection, for m= 2 we introduce new families of relative difference sets in G× Q8, as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.

AB - Let G=Cn1×⋯×Cnm be an abelian group of order n= n1⋯ nm, where each Cnt is cyclic of order nt. We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in G× Q8 relative to the centre Z(Q8) and the perfect arrays of size n1× ⋯ × nm over the quaternionic alphabet Q8∪ qQ8, where q= (1 + i+ j+ k) / 2. In view of this connection, for m= 2 we introduce new families of relative difference sets in G× Q8, as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.

KW - Hadamard matrices

KW - Perfect arrays

KW - Quaternions

KW - Relative difference sets

UR - http://www.scopus.com/inward/record.url?scp=85053554534&partnerID=8YFLogxK

U2 - 10.1007/s11786-018-0376-y

DO - 10.1007/s11786-018-0376-y

M3 - Article

VL - 12

SP - 397

EP - 406

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

SN - 1661-8270

IS - 4

ER -