699 research outputs found
New constructions for covering designs
A {\em covering design}, or {\em covering}, is a family of
-subsets, called blocks, chosen from a -set, such that each -subset is
contained in at least one of the blocks. The number of blocks is the covering's
{\em size}, and the minimum size of such a covering is denoted by .
This paper gives three new methods for constructing good coverings: a greedy
algorithm similar to Conway and Sloane's algorithm for lexicographic
codes~\cite{lex}, and two methods that synthesize new coverings from
preexisting ones. Using these new methods, together with results in the
literature, we build tables of upper bounds on for , , and .
Asymptotically optimal covering designs
A (v,k,t) covering design, or covering, is a family of k-subsets, called
blocks, chosen from a v-set, such that each t-subset is contained in at least
one of the blocks. The number of blocks is the covering's size}, and the
minimum size of such a covering is denoted by C(v,k,t). It is easy to see that
a covering must contain at least (v choose t)/(k choose t) blocks, and in 1985
R\"odl [European J. Combin. 5 (1985), 69-78] proved a long-standing conjecture
of Erd\H{o}s and Hanani [Publ. Math. Debrecen 10 (1963), 10-13] that for fixed
k and t, coverings of size (v choose t)/(k choose t) (1+o(1)) exist (as v \to
\infty).
An earlier paper by the first three authors [J. Combin. Des. 3 (1995),
269-284] gave new methods for constructing good coverings, and gave tables of
upper bounds on C(v,k,t) for small v, k, and t. The present paper shows that
two of those constructions are asymptotically optimal: For fixed k and t, the
size of the coverings constructed matches R\"odl's bound. The paper also makes
the o(1) error bound explicit, and gives some evidence for a much stronger
bound
Controlled Dephasing of a Quantum Dot: From Coherent to Sequential Tunneling
Resonant tunneling through identical potential barriers is a textbook problem
in quantum mechanics. Its solution yields total transparency (100% tunneling)
at discrete energies. This dramatic phenomenon results from coherent
interference among many trajectories, and it is the basis of transport through
periodic structures. Resonant tunneling of electrons is commonly seen in
semiconducting 'quantum dots'. Here we demonstrate that detecting
(distinguishing) electron trajectories in a quantum dot (QD) renders the QD
nearly insulating. We couple trajectories in the QD to a 'detector' by
employing edge channels in the integer quantum Hall regime. That is, we couple
electrons tunneling through an inner channel to electrons in the neighboring
outer, 'detector' channel. A small bias applied to the detector channel
suffices to dephase (quench) the resonant tunneling completely. We derive a
formula for dephasing that agrees well with our data and implies that just a
few electrons passing through the detector channel suffice to dephase the QD
completely. This basic experiment shows how path detection in a QD induces a
transition from delocalization (due to coherent tunneling) to localization
(sequential tunneling)
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