Construction of relative difference sets and Hadamard groups

Bernhard Schmidt, Ming Ming Tan

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


'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 d22e26fi=1s p i4aii=1t qi 2i=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 languageEnglish (US)
Pages (from-to)105-119
Number of pages15
JournalDesigns, Codes, and Cryptography
Issue number1
StatePublished - Oct 2014
Externally publishedYes


  • 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


Dive into the research topics of 'Construction of relative difference sets and Hadamard groups'. Together they form a unique fingerprint.

Cite this