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+16b9c10 d22e26f ∝i=1s p i4ai ∝i=1t qi 2 ∝i=1u ((ri+1)/2)r ivi) ∝i=1w si under the following conditions: a,b,c,d,e,f,s,t,u,w are nonnegative integers, a 1,⋯,ar and v1,⋯,vu are positive integers, p1,⋯,ps are odd primes, q 1,⋯,qt and r1,⋯,ru are prime powers with qi≡ 1 (mod 4) and ri≡ 1 (mod 4) for all i, s1,⋯,sw are integers with 1≤ si ≤ 33 or si∈ {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 ℤ4with 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