A Hadamard difference set (HDS) has the parameters (4N2, 2N2 − N, N2 − 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 × Z2pr contains a (hp2r, √hpr(2√hpr − 1), √hpr(√hpr − 1)) HDS (HDS), p a prime not dividing |H| = h and pj ≡ −1 (mod exp(H)) for some j, then H× Z2pt has a (hp2t, √hpt(2√hpt − 1), √hpt(√hpt − 1)) HDS for every 0⩽t⩽r. Thus, if these families do not exist, we simply need to show that H × Z2p does not support a HDS. We give two examples of families that are ruled out by this procedure.
Copyright © 1997, Elsevier. This article first appeared in Journal of Statistical Planning and Inference: 62:1 (1997), 13-20.
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Davis, James A., and Jonathan Jedwab. Journal of Statistical Planning and Inference 62, no. 1 (July 21, 1997): 13-20. doi: 10.1016/S0378-3758(96)00162-0.