27,576 research outputs found
On existence and construction of balanced arrays
AbstractThis paper surveys existence conditions and construction procedures for balanced arrays. A necessary and sufficient condition for the existence of an s-symbol balanced array of strength t with m constraints is discussed. The construction of an s-symbol balanced array of strength 2, based on an (r, λ)-design with mutually balanced nested subdesigns, is presented. Related open problems are exhibited
Resolvable designs with large blocks
Resolvable designs with two blocks per replicate are studied from an
optimality perspective. Because in practice the number of replicates is
typically less than the number of treatments, arguments can be based on the
dual of the information matrix and consequently given in terms of block
concurrences. Equalizing block concurrences for given block sizes is often, but
not always, the best strategy. Sufficient conditions are established for
various strong optimalities and a detailed study of E-optimality is offered,
including a characterization of the E-optimal class. Optimal designs are found
to correspond to balanced arrays and an affine-like generalization.Comment: Published at http://dx.doi.org/10.1214/009053606000001253 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
Absolutely Maximally Entangled (AME) states are those multipartite quantum
states that carry absolute maximum entanglement in all possible partitions. AME
states are known to play a relevant role in multipartite teleportation, in
quantum secret sharing and they provide the basis novel tensor networks related
to holography. We present alternative constructions of AME states and show
their link with combinatorial designs. We also analyze a key property of AME,
namely their relation to tensors that can be understood as unitary
transformations in every of its bi-partitions. We call this property
multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom
On 1-factorizations of Bipartite Kneser Graphs
It is a challenging open problem to construct an explicit 1-factorization of
the bipartite Kneser graph , which contains as vertices all -element
and -element subsets of and an edge between any
two vertices when one is a subset of the other. In this paper, we propose a new
framework for designing such 1-factorizations, by which we solve a nontrivial
case where and is an odd prime power. We also revisit two classic
constructions for the case --- the \emph{lexical factorization} and
\emph{modular factorization}. We provide their simplified definitions and study
their inner structures. As a result, an optimal algorithm is designed for
computing the lexical factorizations. (An analogous algorithm for the modular
factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a
odd prime powe
Genuinely multipartite entangled states and orthogonal arrays
A pure quantum state of N subsystems with d levels each is called
k-multipartite maximally entangled state, written k-uniform, if all its
reductions to k qudits are maximally mixed. These states form a natural
generalization of N-qudits GHZ states which belong to the class 1-uniform
states. We establish a link between the combinatorial notion of orthogonal
arrays and k-uniform states and prove the existence of several new classes of
such states for N-qudit systems. In particular, known Hadamard matrices allow
us to explicitly construct 2-uniform states for an arbitrary number of N>5
qubits. We show that finding a different class of 2-uniform states would imply
the Hadamard conjecture, so the full classification of 2-uniform states seems
to be currently out of reach. Additionally, single vectors of another class of
2-uniform states are one-to-one related to maximal sets of mutually unbiased
bases. Furthermore, we establish links between existence of k-uniform states,
classical and quantum error correction codes and provide a novel graph
representation for such states.Comment: 24 pages, 7 figures. Comments are very welcome
Lyndon Array Construction during Burrows-Wheeler Inversion
In this paper we present an algorithm to compute the Lyndon array of a string
of length as a byproduct of the inversion of the Burrows-Wheeler
transform of . Our algorithm runs in linear time using only a stack in
addition to the data structures used for Burrows-Wheeler inversion. We compare
our algorithm with two other linear-time algorithms for Lyndon array
construction and show that computing the Burrows-Wheeler transform and then
constructing the Lyndon array is competitive compared to the known approaches.
We also propose a new balanced parenthesis representation for the Lyndon array
that uses bits of space and supports constant time access. This
representation can be built in linear time using words of space, or in
time using asymptotically the same space as
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