21,554 research outputs found
Note on cubature formulae and designs obtained from group orbits
In 1960, Sobolev proved that for a finite reflection group G, a G-invariant
cubature formula is of degree t if and only if it is exact for all G-invariant
polynomials of degree at most t. In this paper, we find some observations on
invariant cubature formulas and Euclidean designs in connection with the
Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998)
on necessary and sufficient conditions for the existence of cubature formulas
with some strong symmetry. The new proof is shorter and simpler compared to the
original one by Xu, and moreover gives a general interpretation of the
analytically-written conditions of Xu's theorems. Second, we extend a theorem
by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean
designs, and thereby classify tight Euclidean designs obtained from unions of
the orbits of the corner vectors. This result generalizes a theorem of Bajnok
(2007) which classifies tight Euclidean designs invariant under the Weyl group
of type B to other finite reflection groups.Comment: 18 pages, no figur
Large Sets of t-Designs
We investigate the existence of large sets of t-designs. We introduce t-wise equivalence
and (n, t)-partitionable sets. We propose a general approach to construct large
sets of t-designs. Then, we consider large sets of a prescribed size n. We partition
the set of all k-subsets of a v-set into several parts, each can be written as product
of two trivial designs. Utilizing these partitions we develop some recursive methods
to construct large sets of t-designs. Then, we direct our attention to the large sets
of prime size. We prove two extension theorems for these large sets. These theorems
are the only known recursive constructions for large sets which do not put any
additional restriction on the parameters, and work for all t and k. One of them,
has even a further advantage; it increase the strength of the large set by one, and it
can be used recursively which makes it one of a kind. Then applying this theorem
recursively, we construct large sets of t-designs for all t and some blocksizes k.
Hartman conjectured that the necessary conditions for the existence of a large
set of size two are also sufficient. We suggest a recursive approach to the Hartman
conjecture, which reduces this conjecture to the case that the blocksize is a power
of two, and the order is very small. Utilizing this approach, we prove the Hartman
conjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitely
many k, and for the rest of them there are at most k/2 exceptions.
In Chapter 4 we consider the case k = t + 1. We modify the recursive methods
developed by Teirlinck, and then we construct some new infinite families of large
sets of t-designs (for all t), some of them are the smallest known large sets. We also
prove that if k = t + 1, then the Hartman conjecture is asymptotically correct.</p
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
Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted
quantum low-density parity-check (LDPC) codes, which is based on combinatorial
design theory. Explicit constructions are given for entanglement-assisted
quantum error-correcting codes (EAQECCs) with many desirable properties. These
properties include the requirement of only one initial entanglement bit, high
error correction performance, high rates, and low decoding complexity. The
proposed method produces infinitely many new codes with a wide variety of
parameters and entanglement requirements. Our framework encompasses various
codes including the previously known entanglement-assisted quantum LDPC codes
having the best error correction performance and many new codes with better
block error rates in simulations over the depolarizing channel. We also
determine important parameters of several well-known classes of quantum and
classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review
Frame difference families and resolvable balanced incomplete block designs
Frame difference families, which can be obtained via a careful use of
cyclotomic conditions attached to strong difference families, play an important
role in direct constructions for resolvable balanced incomplete block designs.
We establish asymptotic existences for several classes of frame difference
families. As corollaries new infinite families of 1-rotational
-RBIBDs over are
derived, and the existence of -RBIBDs is discussed. We construct
-RBIBDs for , whose
existence were previously in doubt. As applications, we establish asymptotic
existences for an infinite family of optimal constant composition codes and an
infinite family of strictly optimal frequency hopping sequences.Comment: arXiv admin note: text overlap with arXiv:1702.0750
New -designs from strong difference families
Strong difference families are an interesting class of discrete structures
which can be used to derive relative difference families. Relative difference
families are closely related to -designs, and have applications in
constructions for many significant codes, such as optical orthogonal codes and
optical orthogonal signature pattern codes. In this paper, with a careful use
of cyclotomic conditions attached to strong difference families, we improve the
lower bound on the asymptotic existence results of -DFs for .
We improve Buratti's existence results for - designs and
- designs, and establish the existence of seven new
- designs for
,
.Comment: Version 1 is named "Improved cyclotomic conditions leading to new
2-designs: the use of strong difference families". Major revision according
to the referees' comment
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
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