42,597 research outputs found
Quantum Random Access Codes with Shared Randomness
We consider a communication method, where the sender encodes n classical bits
into 1 qubit and sends it to the receiver who performs a certain measurement
depending on which of the initial bits must be recovered. This procedure is
called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success
probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no
classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not
possible.
We extend this model with shared randomness (SR) that is accessible to both
parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an
upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79)
QRACs match this upper bound). We discuss some particular constructions for
several small values of n.
We also study the classical counterpart of this model where n bits are
encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal
construction for such codes and find their success probability exactly--it is
less than in the quantum case.
Interactive 3D quantum random access codes are available on-line at
http://home.lanet.lv/~sd20008/racs .Comment: 51 pages, 33 figures. New sections added: 1.2, 3.5, 3.8.2, 5.4 (paper
appears to be shorter because of smaller margins). Submitted as M.Math thesis
at University of Waterloo by M
Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs
This paper presents an extension of the Elias bound on the minimum distance
of codes for discrete alphabets with general, possibly infinite-valued,
distances. The bound is obtained by combining a previous extension of the Elias
bound, introduced by Blahut, with an extension of a bound previously introduced
by the author which builds upon ideas of Gallager, Lov\'asz and Marton. The
result can in fact be interpreted as a unification of the Elias bound and of
Lov\'asz's bound on graph (or zero-error) capacity, both being recovered as
particular cases of the one presented here. Previous extensions of the Elias
bound by Berlekamp, Blahut and Piret are shown to be included as particular
cases of our bound. Applications to the reliability function are then
discussed.Comment: Accepted, IEEE Transaction on Information Theor
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
Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes
A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid
Asymptotic probability bounds on the peak distribution of complex multicarrier signals without Gaussian assumption
Multicarrier signals exhibit a large peak to mean envelope power ratio (PMEPR). In this paper, we derive the lower and upper probability bounds for the PMEPR distribution when entries of the codeword, C, are chosen independently from a symmetric q-ary PSK or QAM constellation, C /spl isin/ /spl Qscr/;/sup nq/, or C is chosen from a complex n dimensional sphere, /spl Omega//sup n/ when the number of subcarriers, n, is large and without any Gaussian assumption on either the joint distribution or any sample of the multicarrier signal. Even though the worst case PMEPR is of the order of n, the main result is that the PMEPR of a random codeword C chosen from /spl Qscr/;/sup nq/ or /spl Omega//sup n/ is log n with probability one, asymptotically. A Varsharmov-Gilbert (VG) style bound for the achievable rate and minimum Hamming distance of codes chosen from /spl Qscr/;/sup nq/, with PMEPR of less than log n is obtained. It is proved that asymptotically, the VG bound remains the same for the codes chosen from /spl Qscr/;/sup nq/ with PMEPR of less than log n
On multicarrier signals where the PMEPR of a random codeword is asymptotically log n
Multicarrier signals exhibit a large peak-to-mean envelope power ratio (PMEPR). In this correspondence, without using a Gaussian assumption, we derive lower and upper probability bounds for the PMEPR distribution when the number of subcarriers n is large. Even though the worst case PMEPR is of the order of n, the main result is that the PMEPR of a random codeword C=(c/sub 1/,...,c/sub n/) is logn with probability approaching one asymptotically, for the following three general cases: i) c/sub i/'s are independent and identically distributed (i.i.d.) chosen from a complex quadrature amplitude modulation (QAM) constellation in which the real and imaginary part of c/sub i/ each has i.i.d. and even distribution (not necessarily uniform), ii) c/sub i/'s are i.i.d. chosen from a phase-shift keying (PSK) constellation where the distribution over the constellation points is invariant under /spl pi//2 rotation, and iii) C is chosen uniformly from a complex sphere of dimension n. Based on this result, it is proved that asymptotically, the Varshamov-Gilbert (VG) bound remains the same for codes with PMEPR of less than logn chosen from QAM/PSK constellations
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