5,559 research outputs found

    A Unifying Construction for Difference Sets

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    We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, λ,n)=(22d+4(22d+2−1)/3, 22d+1(22d+3+1)/3, 22d+1(22d+1+1)/3, 24d+2) for d⩾0. The construction establishes that a McFarland difference set exists in an abelian group of order 22d+3(22d+1+1)/3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order pr. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r\u3e1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups

    Hadamard partitioned difference families and their descendants

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    If DD is a (4u2,2u2−u,u2−u)(4u^2,2u^2-u,u^2-u) Hadamard difference set (HDS) in GG, then {G,G∖D}\{G,G\setminus D\} is clearly a (4u2,[2u2−u,2u2+u],2u2)(4u^2,[2u^2-u,2u^2+u],2u^2) partitioned difference family (PDF). Any (v,K,λ)(v,K,\lambda)-PDF will be said of Hadamard-type if v=2λv=2\lambda as the one above. We present a doubling construction which, starting from any such PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order 4u2(2n+1)4u^2(2n+1) and three block-sizes 4u2−2u4u^2-2u, 4u24u^2 and 4u2+2u4u^2+2u, whenever we have a (4u2,2u2−u,u2−u)(4u^2,2u^2-u,u^2-u)-HDS and the maximal prime power divisors of 2n+12n+1 are all greater than 4u2+2u4u^2+2u

    On quasi-orthogonal cocycles

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    We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.Junta de Andalucía FQM-01
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