1,563 research outputs found
Cyclic division algebras: a tool for space-time coding
Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space–Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank.
Extensive work has been done on Space–Time coding, aiming at
finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to
improve the design of good codes.
The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes
Constructive spherical codes on layers of flat tori
A new class of spherical codes is constructed by selecting a finite subset of
flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing
a structured codebook on each torus layer. The resulting spherical code can be
the image of a lattice restricted to a specific hyperbox in R^L in each layer.
Group structure and homogeneity, useful for efficient storage and decoding, are
inherited from the underlying lattice codebook. A systematic method for
constructing such codes are presented and, as an example, the Leech lattice is
used to construct a spherical code in R^{48}. Upper and lower bounds on the
performance, the asymptotic packing density and a method for decoding are
derived.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information
Theor
Algebraic Approach to Physical-Layer Network Coding
The problem of designing physical-layer network coding (PNC) schemes via
nested lattices is considered. Building on the compute-and-forward (C&F)
relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain
using information-theoretic tools, an algebraic approach is taken to show its
potential in practical, non-asymptotic, settings. A general framework is
developed for studying nested-lattice-based PNC schemes---called lattice
network coding (LNC) schemes for short---by making a direct connection between
C&F and module theory. In particular, a generic LNC scheme is presented that
makes no assumptions on the underlying nested lattice code. C&F is
re-interpreted in this framework, and several generalized constructions of LNC
schemes are given. The generic LNC scheme naturally leads to a linear network
coding channel over modules, based on which non-coherent network coding can be
achieved. Next, performance/complexity tradeoffs of LNC schemes are studied,
with a particular focus on hypercube-shaped LNC schemes. The error probability
of this class of LNC schemes is largely determined by the minimum inter-coset
distances of the underlying nested lattice code. Several illustrative
hypercube-shaped LNC schemes are designed based on Construction A and D,
showing that nominal coding gains of 3 to 7.5 dB can be obtained with
reasonable decoding complexity. Finally, the possibility of decoding multiple
linear combinations is considered and related to the shortest independent
vectors problem. A notion of dominant solutions is developed together with a
suitable lattice-reduction-based algorithm.Comment: Submitted to IEEE Transactions on Information Theory, July 21, 2011.
Revised version submitted Sept. 17, 2012. Final version submitted July 3,
201
Finite-Block-Length Analysis in Classical and Quantum Information Theory
Coding technology is used in several information processing tasks. In
particular, when noise during transmission disturbs communications, coding
technology is employed to protect the information. However, there are two types
of coding technology: coding in classical information theory and coding in
quantum information theory. Although the physical media used to transmit
information ultimately obey quantum mechanics, we need to choose the type of
coding depending on the kind of information device, classical or quantum, that
is being used. In both branches of information theory, there are many elegant
theoretical results under the ideal assumption that an infinitely large system
is available. In a realistic situation, we need to account for finite size
effects. The present paper reviews finite size effects in classical and quantum
information theory with respect to various topics, including applied aspects
Algebraic number theory and code design for Rayleigh fading channels
Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.
Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available.
The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible
to a large audience
- …