# Williamson Matrices and a Conjecture of Ito's

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## Abstract

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4*t*, 2, 4*t*, 2t)-difference sets in the dicyclic groups *Q*_{8t} = 〈*a*, *b*|*a*^{4t} = *b*^{4} = 1, *a*^{2t} = *b*^{2}, *b*^{-1}ab = a^{-1}〉 for all *t* of the form *t* = 2^{a} · 10^{ b } · 26^{ c } · m with *a*, *b*, *c* ≥ 0, m ≡ 1\ (mod 2), whenever 2*m*-1 or 4*m*-1 is a prime power or there is a Williamson matrix over ℤ_{m}. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4*t*, 2, 4*t*, 2*t*)-difference sets in *Q*_{8t} for every positive integer *t*. We also give simpler alternative constructions for relative (4*t*, 2, 4*t*, 2*t*)-difference sets in *Q*_{8t} for all *t* such that 2*t* - 1 or 4*t* - 1 is a prime power. Relative difference sets in *Q*_{8t} with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all *t* ≤ 46.

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## References

- 1.L. D. Baumert, Hadamard matrices of Williamson type,
*Math. Comp.*, Vol. 19 (1965) pp. 442-447.Google Scholar - 2.J. A. Davis and J. Jedwab, A unifying construction of difference sets,
*J. Combin. Theory*, Vol. A 80 (1997) pp. 13-78.Google Scholar - 3.D. Z. Dokovic, Williamson matrices of order 4
*n*for*n*= 33, 35, 39,*Discrete Math.*, Vol. 115 (1993) pp. 267-271.Google Scholar - 4.S. Eliahou, M. Kervaire, B. Saffari, A new restriction on the length of Golay complementary sequences,
*J. Comb. Theory*, Vol. A 55 (1990) pp. 49-59.Google Scholar - 5.D. L. Flannery, Cocyclic Hadamard Matrices and Hadamard Groups are Equivalent,
*J. Algebra*, Vol. 192 (1997) pp. 749-779.Google Scholar - 6.N. Ito, On Hadamard groups,
*J. Algebra*, Vol. 168 (1994) pp. 981-987.Google Scholar - 7.N. Ito, On Hadamard groups, II,
*J. Algebra*, Vol. 169 (1994) pp. 936-942.Google Scholar - 8.N. Ito, Some remarks on Hadamard groups, Groups-Korea 94, de Gruyter, Berlin/New York (1995) pp. 149-155.Google Scholar
- 9.N. Ito, Remarks on Hadamard groups,
*Kyushu J. Math.*, Vol. 50 (1996) pp. 83-91.Google Scholar - 10.N. Ito, Remarks on Hadamard groups, II,
*Rep. Fac. Sci. Technol. Meijo Univ.*, No. 37 (1997) pp. 1-7.Google Scholar - 11.N. Ito, On Hadamard groups III,
*Kyushu J. Math.*, Vol. 51 (1997) pp. 369-379.Google Scholar - 12.A. Pott, Finite geometry and character theory,
*Springer Lecture Notes*, Vol. 1601, New York (1995).Google Scholar - 13.E. Spence, An Infinite Family of Williamson Matrices,
*J. Austral. Math. Soc.*, Vol. 24A (1977) pp. 252-256.Google Scholar - 14.R. J. Turyn, An infinite class of Williamson matrices,
*J. Combin. Theory*, Vol. A 12 (1972) pp. 319-321.Google Scholar - 15.R. J. Turyn, Hadamard Matrices, Baumert-Hall Units, Four-Symbol Sequences, Pulse Compression, and Surface Wave Encodings,
*J. Comb. Theory*, Vol. A 16 (1974) pp. 313-333.Google Scholar - 16.A. L. Whiteman, An infinite family of Hadamard matrices of Williamson type,
*J. Combin. Theory*, Vol. A 14 (1973) pp. 334-340.Google Scholar - 17.J. Williamson, Hadamard's determinant problem and the sum of four squares,
*Duke Math. J.*, Vol. 11 (1944) pp. 65-81.Google Scholar