Nested Hadamard difference sets |
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| Institution: | 1. Department of Mathematics, University of Richmond, Richmond, VA 23173. USA;2. Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK;1. Division of Natural Science, Faculty of Advanced Science and Technology, Kumamoto University, 2-40-1 Kurokami, Kumamoto 860-8555, Japan;2. Department of Mathematics, National Defense Academy of Japan, Yokosuka, Kanagawa 239-8686, Japan;3. Department of Mathematics and National Center for Applied Mathematics Shenzhen, Southern University of Science and Technology, Shenzhen 518055, China |
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| Abstract: | 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. |
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