356 research outputs found
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
Finite Dimensional Infinite Constellations
In the setting of a Gaussian channel without power constraints, proposed by
Poltyrev, the codewords are points in an n-dimensional Euclidean space (an
infinite constellation) and the tradeoff between their density and the error
probability is considered. The capacity in this setting is the highest
achievable normalized log density (NLD) with vanishing error probability. This
capacity as well as error exponent bounds for this setting are known. In this
work we consider the optimal performance achievable in the fixed blocklength
(dimension) regime. We provide two new achievability bounds, and extend the
validity of the sphere bound to finite dimensional infinite constellations. We
also provide asymptotic analysis of the bounds: When the NLD is fixed, we
provide asymptotic expansions for the bounds that are significantly tighter
than the previously known error exponent results. When the error probability is
fixed, we show that as n grows, the gap to capacity is inversely proportional
(up to the first order) to the square-root of n where the proportion constant
is given by the inverse Q-function of the allowed error probability, times the
square root of 1/2. In an analogy to similar result in channel coding, the
dispersion of infinite constellations is 1/2nat^2 per channel use. All our
achievability results use lattices and therefore hold for the maximal error
probability as well. Connections to the error exponent of the power constrained
Gaussian channel and to the volume-to-noise ratio as a figure of merit are
discussed. In addition, we demonstrate the tightness of the results numerically
and compare to state-of-the-art coding schemes.Comment: 54 pages, 13 figures. Submitted to IEEE Transactions on Information
Theor
Lattices from Codes for Harnessing Interference: An Overview and Generalizations
In this paper, using compute-and-forward as an example, we provide an
overview of constructions of lattices from codes that possess the right
algebraic structures for harnessing interference. This includes Construction A,
Construction D, and Construction (previously called product
construction) recently proposed by the authors. We then discuss two
generalizations where the first one is a general construction of lattices named
Construction subsuming the above three constructions as special cases
and the second one is to go beyond principal ideal domains and build lattices
over algebraic integers
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