7,412 research outputs found
Hadamard matrices modulo 5
In this paper we introduce modular symmetric designs and use them to study
the existence of Hadamard matrices modulo 5. We prove that there exist
5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n
!= 6, 11. In particular, this solves the 5-modular version of the Hadamard
conjecture.Comment: 7 pages, submitted to JC
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
Some non-existence and asymptotic existence results for weighing matrices
Orthogonal designs and weighing matrices have many applications in areas such
as coding theory, cryptography, wireless networking and communication. In this
paper, we first show that if positive integer cannot be written as the sum
of three integer squares, then there does not exist any skew-symmetric weighing
matrix of order and weight , where is an odd positive integer. Then
we show that for any square , there is an integer such that for each
, there is a symmetric weighing matrix of order and weight .
Moreover, we improve some of the asymptotic existence results for weighing
matrices obtained by Eades, Geramita and Seberry.Comment: To appear in International Journal of Combinatorics (Hindawi). in
Int. J. Combin. (Feb 2016
More nonexistence results for symmetric pair coverings
A -covering is a pair , where is a
-set of points and is a collection of -subsets of
(called blocks), such that every unordered pair of points in is contained
in at least blocks in . The excess of such a covering is
the multigraph on vertex set in which the edge between vertices and
has multiplicity , where is the number of blocks which
contain the pair . A covering is symmetric if it has the same number
of blocks as points. Bryant et al.(2011) adapted the determinant related
arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the
nonexistence of certain symmetric coverings with -regular excesses. Here, we
adapt the arguments related to rational congruence of matrices and show that
they imply the nonexistence of some cyclic symmetric coverings and of various
symmetric coverings with specified excesses.Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its
Application
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
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