417 research outputs found
Hadamard matrices modulo p and small modular Hadamard matrices
We use modular symmetric designs to study the existence of Hadamard matrices
modulo certain primes. We solve the -modular and -modular versions of
the Hadamard conjecture for all but a finite number of cases. In doing so, we
state a conjecture for a sufficient condition for the existence of a
-modular Hadamard matrix for all but finitely many cases. When is a
primitive root of a prime , we conditionally solve this conjecture and
therefore the -modular version of the Hadamard conjecture for all but
finitely many cases when , and prove a weaker result for
. Finally, we look at constraints on the existence of
-modular Hadamard matrices when the size of the matrix is small compared to
.Comment: 14 pages; to appear in the Journal of Combinatorial Designs; proofs
of Lemma 4.7 and Theorem 5.2 altered in response to referees' comment
A new structure for difference matrices over abelian -groups
A difference matrix over a group is a discrete structure that is intimately
related to many other combinatorial designs, including mutually orthogonal
Latin squares, orthogonal arrays, and transversal designs. Interest in
constructing difference matrices over -groups has been renewed by the recent
discovery that these matrices can be used to construct large linking systems of
difference sets, which in turn provide examples of systems of linked symmetric
designs and association schemes. We survey the main constructive and
nonexistence results for difference matrices, beginning with a classical
construction based on the properties of a finite field. We then introduce the
concept of a contracted difference matrix, which generates a much larger
difference matrix. We show that several of the main constructive results for
difference matrices over abelian -groups can be substantially simplified and
extended using contracted difference matrices. In particular, we obtain new
linking systems of difference sets of size in infinite families of abelian
-groups, whereas previously the largest known size was .Comment: 27 pages. Discussion of new reference [LT04
Cocyclic simplex codes of type alpha over Z4 and Z2s
Over the past decade, cocycles have been used to construct Hadamard and generalized Hadamard matrices. This, in turn, has led to the construction of codes-self-dual and others. Here we explore these ideas further to construct cocyclic complex and Butson-Hadamard matrices, and subsequently we use the matrices to construct simplex codes of type /spl alpha/ over Z(4) and Z(2/sup s/), respectively
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