Nested Hadamard difference sets

Abstract

A Hadamard difference set (HDS) has the parameters (4 N 2, 2 N 2 − N, N 2 − N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. We show that if a group of the form H × Z 2 p r contains a ( hp 2 r , √ hp r (2√ hp r − 1), √ hp r (√ hp r − 1)) HDS (HDS), p a prime not dividing | H| = h and p j ≡ −1 (mod exp( H)) for some j, then H × Z 2 p t has a ( hp 2 t , √ hp t (2√ hp t − 1), √ hp t (√ hp t − 1)) HDS for every 0⩽ t⩽ r. Thus, if these families do not exist, we simply need to show that H × Z 2 p does not support a HDS. We give two examples of families that are ruled out by this procedure.