3,393 research outputs found
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
On Bounded Weight Codes
The maximum size of a binary code is studied as a function of its length N,
minimum distance D, and minimum codeword weight W. This function B(N,D,W) is
first characterized in terms of its exponential growth rate in the limit as N
tends to infinity for fixed d=D/N and w=W/N. The exponential growth rate of
B(N,D,W) is shown to be equal to the exponential growth rate of A(N,D) for w <=
1/2, and equal to the exponential growth rate of A(N,D,W) for 1/2< w <= 1.
Second, analytic and numerical upper bounds on B(N,D,W) are derived using the
semidefinite programming (SDP) method. These bounds yield a non-asymptotic
improvement of the second Johnson bound and are tight for certain values of the
parameters
Semidefinite programming converse bounds for quantum communication
We derive several efficiently computable converse bounds for quantum
communication over quantum channels in both the one-shot and asymptotic regime.
First, we derive one-shot semidefinite programming (SDP) converse bounds on the
amount of quantum information that can be transmitted over a single use of a
quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes,
Nat. Commun. 7, 2016]. As applications, we study quantum communication over
depolarizing channels and amplitude damping channels with finite resources.
Second, we find an SDP strong converse bound for the quantum capacity of an
arbitrary quantum channel, which means the fidelity of any sequence of codes
with a rate exceeding this bound will vanish exponentially fast as the number
of channel uses increases. Furthermore, we prove that the SDP strong converse
bound improves the partial transposition bound introduced by Holevo and Werner.
Third, we prove that this SDP strong converse bound is equal to the so-called
max-Rains information, which is an analog to the Rains information introduced
in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP
strong converse bound is weaker than the Rains information, but it is
efficiently computable for general quantum channels.Comment: 17 pages, extended version of arXiv:1601.06888. v3 is closed to the
published version, IEEE Transactions on Information Theory, 201
Semidefinite programming, harmonic analysis and coding theory
These lecture notes where presented as a course of the CIMPA summer school in
Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
This version is an update June 2010
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
General Rank Multiuser Downlink Beamforming With Shaping Constraints Using Real-valued OSTBC
In this paper we consider optimal multiuser downlink beamforming in the
presence of a massive number of arbitrary quadratic shaping constraints. We
combine beamforming with full-rate high dimensional real-valued orthogonal
space time block coding (OSTBC) to increase the number of beamforming weight
vectors and associated degrees of freedom in the beamformer design. The
original multi-constraint beamforming problem is converted into a convex
optimization problem using semidefinite relaxation (SDR) which can be solved
efficiently. In contrast to conventional (rank-one) beamforming approaches in
which an optimal beamforming solution can be obtained only when the SDR
solution (after rank reduction) exhibits the rank-one property, in our approach
optimality is guaranteed when a rank of eight is not exceeded. We show that our
approach can incorporate up to 79 additional shaping constraints for which an
optimal beamforming solution is guaranteed as compared to a maximum of two
additional constraints that bound the conventional rank-one downlink
beamforming designs. Simulation results demonstrate the flexibility of our
proposed beamformer design
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