Algebraic Number Theory and Code Design for Rayleigh Fading Channels

Oggier, Frédérique; Viterbo, Emanuele

doi:10.1561/0100000003

Cited by 102 publications

(167 citation statements)

References 29 publications

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“…This can be estimated using the infinite lattice constellation and (2.13). Using the theory of algebraic and ideal lattices [3] we find d 2 p,min = 4/25 for the algebraic lattice constellation and d 2 p,min = 1/5 for the algebraically rotated 16-QAM [43]. The performance of the codeword error probability for the three codes is shown in Figure 2 corresponding to a diversity one, while the other two have diversity two.…”

Section: Constellation Shapingmentioning

confidence: 97%

“…Note that d 2 p,min is invariant by translations of the constellation and scales with E 2 s . Full rank diagonal STBCs correspond to full modulation diversity constellations (i.e., no two points have one or more coordinates in common) [43]. We will use these three codes to illustrate the trade-off among diversity, coding gain and constellation shaping.…”

Section: Constellation Shapingmentioning

confidence: 99%

“…Intuitively, the diagonal matrix is designed to skew the constellation in Figure 2.4(b) into the cubic shaped one in Figure 2.5 without loosing the full diversity (see [43]). …”

Section: Constellation Shapingmentioning

confidence: 99%

“…The interested reader can refer to [3,4]. A self contained introduction to algebraic lattices can also be found in [43].…”

Section: Algebraic Latticesmentioning

confidence: 99%

“…, σ n are relative embeddings of L/K, i.e., σ i fixes K for all i. Recall [11,43] that a lattice Λ can be expressed by means of its generator matrix M…”

Section: Algebraic Latticesmentioning

confidence: 99%

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Cyclic Division Algebras: A Tool for Space-Time Coding

Oggier

Belfiore

Viterbo

2007

FNT in Communications and Information Theory

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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.

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Section: Constellation Shapingmentioning

confidence: 97%

Section: Constellation Shapingmentioning

confidence: 99%

“…Intuitively, the diagonal matrix is designed to skew the constellation in Figure 2.4(b) into the cubic shaped one in Figure 2.5 without loosing the full diversity (see [43]). …”

Section: Constellation Shapingmentioning

confidence: 99%

“…The interested reader can refer to [3,4]. A self contained introduction to algebraic lattices can also be found in [43].…”

Section: Algebraic Latticesmentioning

confidence: 99%

“…, σ n are relative embeddings of L/K, i.e., σ i fixes K for all i. Recall [11,43] that a lattice Λ can be expressed by means of its generator matrix M…”

Section: Algebraic Latticesmentioning

confidence: 99%

See 3 more Smart Citations

Cyclic Division Algebras: A Tool for Space-Time Coding

Oggier

Belfiore

Viterbo

2007

FNT in Communications and Information Theory

Self Cite

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show abstract

Ideal Lattices

Costa¹,

Oggier²,

Campello³

et al. 2017

Lattices Applied to Coding for Reliable and Secure Communications

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Lattice Codes for the Gaussian Wiretap Channel

Belfiore

Oggier

Solé

2011

Lecture Notes in Computer Science

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We consider the Gaussian wiretap channel, where two legitimate players Alice and Bob communicate over an additive white Gaussian noise (AWGN) channel, while Eve is eavesdropping, also through an AWGN channel. We propose a coding strategy based on lattice coset encoding. We analyze Eve's probability of decoding, from which we define the secrecy gain as a design criterion for wiretap lattice codes, expressed in terms of the lattice theta series, which characterizes Eve's confusion as a function of the channel parameters. The secrecy gain is studied for even unimodular lattices, and an asymptotic analysis shows that it grows exponentially in the dimension of the lattice. Examples of wiretap lattice codes are given. Interestingly, minimizing Eve's probability of error involves the same optimization of the theta series as does the flatness factor, another newly defined code design that characterizes lattice codes that achieve strong secrecy.

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Algebraic Number Theory and Code Design for Rayleigh Fading Channels

Cited by 102 publications

References 29 publications

Cyclic Division Algebras: A Tool for Space-Time Coding

Cyclic Division Algebras: A Tool for Space-Time Coding

Ideal Lattices

Lattice Codes for the Gaussian Wiretap Channel

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