543 research outputs found
On the base sequence conjecture
Let BS(m,n) denote the set of base sequences (A;B;C;D), with A and B of
length m and C and D of length n. The base sequence conjecture (BSC) asserts
that BS(n+1,n) exist (i.e., are non-empty) for all n. This is known to be true
for n <= 36 and when n is a Golay number. We show that it is also true for n=37
and n=38. It is worth pointing out that BSC is stronger than the famous
Hadamard matrix conjecture. In order to demonstrate the abundance of base
sequences, we have previously attached to BS(n+1,n) a graph Gamma_n and
computed the Gamma_n for n <= 27. We now extend these computations and
determine the Gamma_n for n=28,...,35. We also propose a conjecture describing
these graphs in general.Comment: 19 pages, 10 tables. To appear in Discrete Mathematics
Generalization of Mirsky's theorem on diagonals and eigenvalues of matrices
Mirsky proved that, for the existence of a complex matrix with given
eigenvalues and diagonal entries, the obvious necessary condition is also
sufficient. We generalize this theorem to matrices over any field and provide a
short proof. Moreover, we show that there is a unique companion-matrix-type
solution for this problem.Comment: 3 page
Small orders of Hadamard matrices and base sequences
We update the list of odd integers n<10000 for which an Hadamard matrix of
order 4n is known to exist. We also exhibit the first example of base sequences
BS(40,39). Consequently, there exist T-sequences TS(n) of length n=79. The
first undecided case has the length n=97.Comment: 7 page
Supplementary difference sets with symmetry for Hadamard matrices
First we give an overview of the known supplementary difference sets (SDS)
(A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is
either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson
matrices over the elementary abelian groups of order 25, 27 and 49 are
constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127
are presented. The last of these is obtained from a (127,57,76)-difference
family that we have constructed. An old non-published example of G-matrices of
order 37 is also included.Comment: 16 pages, 2 tables. A few minor changes are made. The paper will
appear in Operators and Matrice
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