Nested Hadamard difference sets

✍ Scribed by James A. Davis; Jonathan Jedwab

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 427 KB
• Volume: 62
• Category: Article
• ISSN: 0378-3758
• DOI: 10.1016/s0378-3758(96)00162-0

✦ Synopsis

A Hadamard difference set (HDS) has the parameters (4N 2, 2N 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 2 group of the form H x Zp, contains a (hp 2r, x/hp~(2x/hp r -1 ), x/hpr(x/hp ~ -1 )) HDS (HDS), p aprime not dividing ]H I =h and pJ~-I (rood exp(H)) for some j, then H xZ~, has a (hp2',x/hp'(2x/hp 'l),x/hpt(x/hp t-1)) HDS for every O<~t<~r. Thus, if these families do not exist, we simply need to show that H × Z~ does not support a HDS. We give two examples of families that are ruled out by this procedure.