## Abstract

'There exist normal (2m,2,2m,m) relative difference sets and thus Hadamard groups of order 4m for all m of the form m= x2 ^{a+t+u+w+δ-ε+1}6^{b}9^{c}10 ^{d}22^{e}26^{f} ∝_{i=1}^{s} p _{i}^{4ai} ∝_{i=1}^{t} q_{i} ^{2} ∝_{i=1}^{u} ((r_{i}+1)/2)r _{i}^{vi}) ∝_{i=1}^{w} s_{i} under the following conditions: a,b,c,d,e,f,s,t,u,w are nonnegative integers, a _{1},⋯,a_{r} and v_{1},⋯,v_{u} are positive integers, p_{1},⋯,p_{s} are odd primes, q _{1},⋯,q_{t} and r_{1},⋯,r_{u} are prime powers with q_{i}≡ 1 (mod 4) and r_{i}≡ 1 (mod 4) for all i, s_{1},⋯,s_{w} are integers with 1≤ s_{i} ≤ 33 or s_{i}∈ {39,43\} for all i, x is a positive integer such that 2x-1 or 4x-1 is a prime power. Moreover, δ =1 if x>1 and c+s>0, δ =0 otherwise, ε=1 if x=1, c + s = 0, and t+u+w>0,ε=0 otherwise. We also obtain some necessary conditions for the existence of (2m,2,2m,m) relative difference sets in partial semidirect products of ℤ_{4}with abelian groups, and provide a table cases for which m≤100 and the existence of such relative difference sets is open.

Original language | English (US) |
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Pages (from-to) | 105-119 |

Number of pages | 15 |

Journal | Designs, Codes, and Cryptography |

Volume | 73 |

Issue number | 1 |

DOIs | |

State | Published - Oct 2014 |

Externally published | Yes |

## Keywords

- Golay sequences
- Hadamard groups
- Semiregular relative difference sets
- Williamson matrices

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Applied Mathematics