6,238 research outputs found
An asymptotic existence result on compressed sensing matrices
For any rational number and all sufficiently large we give a
deterministic construction for an compressed
sensing matrix with -recoverability where . Our
method uses pairwise balanced designs and complex Hadamard matrices in the
construction of -equiangular frames, which we introduce as a
generalisation of equiangular tight frames. The method is general and produces
good compressed sensing matrices from any appropriately chosen pairwise
balanced design. The -recoverability performance is specified as a
simple function of the parameters of the design. To obtain our asymptotic
existence result we prove new results on the existence of pairwise balanced
designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201
Difference Balanced Functions and Their Generalized Difference Sets
Difference balanced functions from to are closely related
to combinatorial designs and naturally define -ary sequences with the ideal
two-level autocorrelation. In the literature, all existing such functions are
associated with the -homogeneous property, and it was conjectured by Gong
and Song that difference balanced functions must be -homogeneous. First we
characterize difference balanced functions by generalized difference sets with
respect to two exceptional subgroups. We then derive several necessary and
sufficient conditions for -homogeneous difference balanced functions. In
particular, we reveal an unexpected equivalence between the -homogeneous
property and multipliers of generalized difference sets. By determining these
multipliers, we prove the Gong-Song conjecture for prime. Furthermore, we
show that every difference balanced function must be balanced or an affine
shift of a balanced function.Comment: 17 page
Resolvable Mendelsohn designs and finite Frobenius groups
We prove the existence and give constructions of a -fold perfect
resolvable -Mendelsohn design for any integers with such that there exists a finite Frobenius group whose kernel
has order and whose complement contains an element of order ,
where is the least prime factor of . Such a design admits as a group of automorphisms and is perfect when is a
prime. As an application we prove that for any integer in prime factorization, and any prime dividing
for , there exists a resolvable perfect -Mendelsohn design that admits a Frobenius group as a group of
automorphisms. We also prove that, if is even and divides for
, then there are at least resolvable -Mendelsohn designs that admit a Frobenius group as a group of
automorphisms, where is Euler's totient function.Comment: Final versio
- β¦