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Spacetime codes for high data rate wireless communication: Performance criterion and code construction
 IEEE TRANS. INFORM. THEORY
, 1998
"... We consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas. Data is encoded by a channel code and the encoded data is split into n streams that are simultaneously transmitted using n transmit ant ..."
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Cited by 1782 (28 self)
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We consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas. Data is encoded by a channel code and the encoded data is split into n streams that are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. We derive performance criteria for designing such codes under the assumption that the fading is slow and frequency nonselective. Performance is shown to be determined by matrices constructed from pairs of distinct code sequences. The minimum rank among these matrices quantifies the diversity gain, while the minimum determinant of these matrices quantifies the coding gain. The results are then extended to fast fading channels. The design criteria are used to design trellis codes for high data rate wireless communication. The encoding/decoding complexity of these codes is comparable to trellis codes employed in practice over Gaussian channels. The codes constructed here provide the best tradeoff between data rate, diversity advantage, and trellis complexity. Simulation results are provided for 4 and 8 PSK signal sets with data rates of 2 and 3 bits/symbol, demonstrating excellent performance that is within 2–3 dB of the outage capacity for these channels using only 64 state encoders.
Multilevel Codes: Theoretical Concepts and Practical Design Rules
, 1999
"... This paper deals with 2 ` ary transmission using multilevel coding (MLC) and multistage decoding (MSD). The known result that MLC and MSD suffice to approach capacity if the rates at each level are appropriately chosen is reviewed. Using multiuser information theory, it is shown that there is a ..."
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Cited by 207 (33 self)
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This paper deals with 2 ` ary transmission using multilevel coding (MLC) and multistage decoding (MSD). The known result that MLC and MSD suffice to approach capacity if the rates at each level are appropriately chosen is reviewed. Using multiuser information theory, it is shown that there is a large space of rate combinations such that MLC and full maximumlikelihood decoding (MLD) can approach capacity. It is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of MLD. The capacity rule, the balanced distances rules, and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design. Simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies. Moreover, topics relevant in practical applications such as signal set labeling, dimensional...
Averaging bounds for lattices and linear codes
 IEEE Trans. Information Theory
, 1997
"... Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofa ..."
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Cited by 99 (1 self)
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Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofasimple lemma for linear codes over GF (p) used with plevel amplitude modulation. The relation between the combinatorial packing of solid bodies and the informationtheoretic “soft packing ” with arbitrarily small, but positive, overlap is illuminated. The “softpacking” results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda–Poltyrev result that spherically shaped lattice codes and adecoder that is unaware of the shaping can achieve the rate 1=2 log2 (P=N).
Optimal nonuniform signaling for Gaussian channels
 IEEE TRANS. INF. THEORY
, 1993
"... Variablerate data transmission schemes in which constellation points are selected according to a nonuniform probability distribution are studied. When the criterion is one of minimizing the average transmitted energy for a given average bit rate, the best possible distribution with which to selec ..."
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Cited by 38 (2 self)
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Variablerate data transmission schemes in which constellation points are selected according to a nonuniform probability distribution are studied. When the criterion is one of minimizing the average transmitted energy for a given average bit rate, the best possible distribution with which to select constellation points is a MaxwellBoltzmann distribution. In principle, when constellation points are selected according to a MaxwellBoltzmann distribution, the ultimate shaping gain (7re/6 or 1.53 dB) can be achieved in any dimension. Nonuniform signaling schemes can be designed by mapping simple variablelength prefix codes onto the constellation. Using the Huffman procedure, prefix codes can be designed that approach the optimal performance. These schemes provide a fixedrate primary channel and a variablerate secondary channel, and are easily incorporated into standard latticetype coded modulation schemes.
Lowdensity lattice codes
 IEEE Transactions on Information Theory
, 2008
"... Abstract—Lowdensity lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the ndimensional Euclidean space as a linear transformation of a correspondi ..."
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Cited by 36 (2 self)
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Abstract—Lowdensity lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the ndimensional Euclidean space as a linear transformation of a corresponding integer message vector b, i.e., x = Gb, where H = G01 is restricted to be sparse. The fact that H is sparse is utilized to develop a lineartime iterative decoding scheme which attains, as demonstrated by simulations, good error performance within 0.5 dB from capacity at block length of n =100,000 symbols. The paper also discusses convergence results and implementation considerations. Index Terms—Iterative decoding, lattice codes, lattices, lowdensity paritycheck (LDPC) code. I.
The Cell Structures of Certain Lattices
, 1991
"... . The most important lattices in Euclidean space of dimension n 8 are the lattices A n (n ³ 2), D n (n ³ 4), E n (n = 6 , 7 , 8) and their duals. In this paper we determine the cell structures of all these lattices and their Voronoi and Delaunay polytopes in a uniform manner. The results for E 6 * ..."
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Cited by 24 (6 self)
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. The most important lattices in Euclidean space of dimension n 8 are the lattices A n (n ³ 2), D n (n ³ 4), E n (n = 6 , 7 , 8) and their duals. In this paper we determine the cell structures of all these lattices and their Voronoi and Delaunay polytopes in a uniform manner. The results for E 6 * and E 7 * simplify recent work of Worley, and also provide what may be new spacefilling polytopes in dimensions 6 and 7. 1. Introduction The CoxeterDynkin diagrams of types A n , D n , E 6 , E 7 and E 8 arise in surprisingly different parts of mathematics  see the discussions by Arnold [1] and Hazewinkel et al. [30]. In the present paper we study __________________ * This paper appeared in {\m Miscellanea mathematica}, P. Hilton, F. Hirzebruch, and R. Remmert, Eds., SpringerVerlag, NY, 1991, pp. 71107. (**) From the English version AutodaFe(Continuum, New York, p. 385) as translated by C. V. Wedgwood: "You have but to know an object by its proper name for it to lose its dange...
The Art of Signaling: Fifty Years of Coding Theory
, 1998
"... In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaus ..."
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Cited by 20 (0 self)
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In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaussian channel have been discovered that come close to these fundamental limits. There is now a substantial algebraic theory of errorcorrecting codes with as many connections to mathematics as to engineering practice, and the last 20 years have seen the construction of algebraicgeometry codes that can be encoded and decoded in polynomial time, and that beat the Gilbert–Varshamov bound. Given the size of coding theory as a subject, this review is of necessity a personal perspective, and the focus is reliable communication, and not source coding or cryptography. The emphasis is on connecting coding theories for Hamming and Euclidean space and on future challenges, specifically in data networking, wireless communication, and quantum information theory.
Performance evaluation of trelliscoded modulation schemes
 Proc. IEEE
, 1994
"... A description of the algorithms to evaluate the main parameters determining the pegormance of a TrellisCoded Modulation (TCM) scheme is presented. TCM schemes are divided into classes that have an increasing degree of symmetry, so as to properly match the various algorithms to each class. The algo ..."
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Cited by 19 (1 self)
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A description of the algorithms to evaluate the main parameters determining the pegormance of a TrellisCoded Modulation (TCM) scheme is presented. TCM schemes are divided into classes that have an increasing degree of symmetry, so as to properly match the various algorithms to each class. The algorithms are compared in terms of computational complexity and tested on a set of multidimensional PSK codes. I.
Golden SpaceTime Trellis Coded Modulation
"... In this paper, we present a multidimensional trellis coded modulation scheme for a high rate 2 × 2 multipleinput multipleoutput (MIMO) system over slow fading channels. Set partitioning of the Golden code [2] is designed specifically to increase the minimum determinant. The branches of the outer t ..."
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Cited by 17 (12 self)
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In this paper, we present a multidimensional trellis coded modulation scheme for a high rate 2 × 2 multipleinput multipleoutput (MIMO) system over slow fading channels. Set partitioning of the Golden code [2] is designed specifically to increase the minimum determinant. The branches of the outer trellis code are labeled with these partitions. Viterbi algorithm is applied for trellis decoding. In order to compute the branch metrics a sphere decoder is used. The general framework for code optimization is given. Performance of the proposed scheme is evaluated by simulation and it is shown that it achieves significant performance gains over uncoded Golden code. 1.
Low Density Lattice Codes
"... Abstract — Low density lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the ndimensional Euclidean space as a linear transformation of a correspon ..."
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Cited by 10 (1 self)
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Abstract — Low density lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the ndimensional Euclidean space as a linear transformation of a corresponding integer message vector b, i.e., x = Gb, where H = G −1 is restricted to be sparse. The fact that H is sparse is utilized to develop a lineartime iterative decoding scheme which attains, as demonstrated by simulations, good error performance within ∼ 0.5dB from capacity at block length of n = 100, 000 symbols. The paper also discusses convergence results and implementation considerations. I.