5,559 research outputs found
A Unifying Construction for Difference Sets
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
If is a Hadamard difference set (HDS) in , then
is clearly a partitioned
difference family (PDF). Any -PDF will be said of Hadamard-type
if 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 and three
block-sizes , and , whenever we have a
-HDS and the maximal prime power divisors of are
all greater than
On quasi-orthogonal cocycles
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 AndalucÃa FQM-01
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