arXiv subjects

Jean-Claude Belfiore

Publications and source records attributed to Jean-Claude Belfiore.

At least 37 records · Page 2Linked to original sources

arXiv2013

Lattice Codes for the Wiretap Gaussian Channel: Construction and Analysis

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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arXiv2013

Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices

In this paper, we address the design of high spectral-efficiency Barnes-Wall (BW) lattice codes which are amenable to low-complexity decoding in additive white Gaussian noise (AWGN) channels. We propose a new method of constructing complex BW lattice codes from linear codes over polynomial rings, and show that the proposed construction provides an explicit method of bit-labeling complex BW lattice codes. To decode the code, we adapt the low-complexity sequential BW lattice decoder (SBWD) recently proposed by Micciancio and Nicolosi. First, we study the error performance of SBWD in decoding the infinite lattice, wherein we analyze the noise statistics in the algorithm, and propose a new upper bound on its error performance. We show that the SBWD is powerful in making correct decisions well beyond the packing radius. Subsequently, we use the SBWD to decode lattice codes through a novel noise-trimming technique. This is the first work that showcases the error performance of SBWD in decoding BW lattice codes of large block lengths.

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arXiv2011

Wyner-Ziv Coding Based on Multidimensional Nested Lattices

Distributed source coding (DSC) addresses the compression of correlated sources without communication links among them. This paper is concerned with the Wyner-Ziv problem: coding of an information source with side information available only at the decoder in the form of a noisy version of the source. Both the theoretical analysis and code design are addressed in the framework of multi-dimensional nested lattice coding (NLC). For theoretical analysis, accurate computation of the rate-distortion function is given under the high-resolution assumption, and a new upper bound using the derivative of the theta series is derived. For practical code design, several techniques with low complexity are proposed. Compared to the existing Slepian-Wolf coded nested quantization (SWC-NQ) for Wyner-Ziv coding based on one or two-dimensional lattices, our proposed multi-dimensional NLC can offer better performance at arguably lower complexity, since it does not require the second stage of Slepian-Wolf coding.

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arXiv2011

Constructive spherical codes near the Shannon bound

Shannon gave a lower bound in 1959 on the binary rate of spherical codes of given minimum Euclidean distance $ρ$. Using nonconstructive codes over a finite alphabet, we give a lower bound that is weaker but very close for small values of $ρ$. The construction is based on the Yaglom map combined with some finite sphere packings obtained from nonconstructive codes for the Euclidean metric. Concatenating geometric codes meeting the TVZ bound with a Lee metric BCH code over $GF(p),$ we obtain spherical codes that are polynomial time constructible. Their parameters outperform those obtained by Lachaud and Stern in 1994. At very high rate they are above 98 per cent of the Shannon bound.

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arXiv2011

The Compute-and-Forward Protocol: Implementation and Practical Aspects

In a recent work, Nazer and Gastpar proposed the Compute-and-Forward strategy as a physical-layer network coding scheme. They described a code structure based on nested lattices whose algebraic structure makes the scheme reliable and efficient. In this work, we consider the implementation of their scheme for real Gaussian channels and one dimensional lattices. We relate the maximization of the transmission rate to the lattice shortest vector problem. We explicit, in this case, the maximum likelihood criterion and show that it can be implemented by using an Inhomogeneous Diophantine Approximation algorithm.

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arXiv2010

Lattice Code Design for the Rayleigh Fading Wiretap Channel

It has been shown recently that coding for the Gaussian Wiretap Channel can be done with nested lattices. A fine lattice intended to the legitimate user must be designed as a usual lattice code for the Gaussian Channel, while a coarse lattice is added to introduce confusion at the eavesdropper, whose theta series must be minimized. We present a design criterion for both the fine and coarse lattice to obtain wiretap lattice codes for the Rayleigh fading Wiretap Channel.

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arXiv2010

Construction of New Delay-Tolerant Space-Time Codes

Perfect Space-Time Codes (STC) are optimal codes in their original construction for Multiple Input Multiple Output (MIMO) systems. Based on Cyclic Division Algebras (CDA), they are full-rate, full-diversity codes, have Non-Vanishing Determinants (NVD) and hence achieve Diversity-Multiplexing Tradeoff (DMT). In addition, these codes have led to optimal distributed space-time codes when applied in cooperative networks under the assumption of perfect synchronization between relays. However, they loose their diversity when delays are introduced and thus are not delay-tolerant. In this paper, using the cyclic division algebras of perfect codes, we construct new codes that maintain the same properties as perfect codes in the synchronous case. Moreover, these codes preserve their full-diversity in asynchronous transmission.

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arXiv2010

A Type II lattice of norm 8 in dimension 72

A Type II lattice of norm 8 in dimension 72 is obtained by Construction A applied to an extended Quadratic Residue code over Z8. Its automorphism group contains a subgroup isomorphic to PSL(2,71).

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arXiv2010

How to Achieve the Optimal DMT of Selective Fading MIMO Channels?

In this paper, we consider a particular class of selective fading channel corresponding to a channel that is selective either in time or in frequency. For this class of channel, we propose a systematic way to achieve the optimal DMT derived in Coronel and Bölcskei, IEEE ISIT, 2007 by extending the non-vanishing determinant (NVD) criterion to the selective channel case. A new code construction based on split NVD parallel codes is then proposed to satisfy the NVD parallel criterion. This result is of significant interest not only in its own right, but also because it settles a long-standing debate in the literature related to the optimal DMT of selective fading channels.

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arXiv2010

Codes over Matrix Rings for Space-Time Coded Modulations

It is known that, for transmission over quasi-static MIMO fading channels with n transmit antennas, diversity can be obtained by using an inner fully diverse space-time block code while coding gain, derived from the determinant criterion, comes from an appropriate outer code. When the inner code has a cyclic algebra structure over a number field, as for perfect space-time codes, an outer code can be designed via coset coding. More precisely, we take the quotient of the algebra by a two-sided ideal which leads to a finite alphabet for the outer code, with a cyclic algebra structure over a finite field or a finite ring. We show that the determinant criterion induces various metrics on the outer code, such as the Hamming and Bachoc distances. When n=2, partitioning the 2x2 Golden code by using an ideal above the prime 2 leads to consider codes over either M2(F_2) or M2(F_2[i]), both being non-commutative alphabets. Matrix rings of higher dimension, suitable for 3x3 and 4x4 perfect codes, give rise to more complex examples.

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arXiv2010

Secrecy Gain: a Wiretap Lattice Code Design

We propose the notion of secrecy gain as a code design criterion for wiretap lattice codes to be used over an additive white Gaussian noise channel. Our analysis relies on the error probabilites of both the legitimate user and the eavesdropper. We focus on geometrical properties of lattices, described by their theta series, to characterize good wiretap codes.

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arXiv2010

Unimodular Lattices for the Gaussian Wiretap Channel

In a recent paper, the authors introduced a lattice invariant called "Secrecy Gain" which measures the confusion experienced by a passive eavesdropper on the Gaussian Wiretap Channel. We study, here, the behavior of this invariant for unimodular lattices by using tools from Modular Forms and show that, for some families of unimodular lattices, indexed by the dimension, the secrecy gain exponentially goes to infinity with the dimension.

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arXiv2010

Outage Efficient Strategies for Network MIMO with Partial CSIT

We consider a multi-cell MIMO downlink (network MIMO) where $B$ base-stations (BS) with $M$ antennas connected to a central station (CS) serve $K$ single-antenna user terminals (UT). Although many works have shown the potential benefits of network MIMO, the conclusion critically depends on the underlying assumptions such as channel state information at transmitters (CSIT) and backhaul links. In this paper, by focusing on the impact of partial CSIT, we propose an outage-efficient strategy. Namely, with side information of all UT's messages and local CSIT, each BS applies zero-forcing (ZF) beamforming in a distributed manner. For a small number of UTs ($K\leq M$), the ZF beamforming creates $K$ parallel MISO channels. Based on the statistical knowledge of these parallel channels, the CS performs a robust power allocation that simultaneously minimizes the outage probability of all UTs and achieves a diversity gain of $B(M-K+1)$ per UT. With a large number of UTs ($K \geq M$), we propose a so-called distributed diversity scheduling (DDS) scheme to select a subset of $\Ks$ UTs with limited backhaul communication. It is proved that DDS achieves a diversity gain of $B\frac{K}{\Ks}(M-\Ks+1)$, which scales optimally with the number of cooperative BSs $B$ as well as UTs. Numerical results confirm that even under realistic assumptions such as partial CSIT and limited backhaul communications, network MIMO can offer high data rates with a sufficient reliability to individual UTs.

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arXiv2010

Augmented Lattice Reduction for MIMO decoding

Lattice reduction algorithms, such as the LLL algorithm, have been proposed as preprocessing tools in order to enhance the performance of suboptimal receivers in MIMO communications. In this paper we introduce a new kind of lattice reduction-aided decoding technique, called augmented lattice reduction, which recovers the transmitted vector directly from the change of basis matrix, and therefore doesn't entail the computation of the pseudo-inverse of the channel matrix or its QR decomposition. We prove that augmented lattice reduction attains the maximum receive diversity order of the channel; simulation results evidence that it significantly outperforms LLL-SIC detection without entailing any additional complexity. A theoretical bound on the complexity is also derived.

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arXiv2008

Algebraic reduction for space-time codes based on quaternion algebras

In this paper we introduce a new right preprocessing method for the decoding of 2x2 algebraic STBCs, called algebraic reduction, which exploits the multiplicative structure of the code. The principle of the new reduction is to absorb part of the channel into the code, by approximating the channel matrix with an element of the maximal order of the algebra. We prove that algebraic reduction attains the receive diversity when followed by a simple ZF detection. Simulation results for the Golden Code show that using MMSE-GDFE left preprocessing, algebraic reduction with simple ZF detection has a loss of only $3 \dB$ with respect to ML decoding.

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arXiv2008

DMT of weighted Parallel Channels: Application to Broadcast Channel

In a broadcast channel with random packet arrival and transmission queues, the stability of the system is achieved by maximizing a weighted sum rate capacity with suitable weights that depend on the queue size. The weighted sum rate capacity using Dirty Paper Coding (DPC) and Zero Forcing (ZF) is asymptotically equivalent to the weighted sum capacity over parallel single-channels. In this paper, we study the Diversity Multiplexing Tradeoff (DMT) of the fading broadcast channel under a fixed weighted sum rate capacity constraint. The DMT of both identical and different parallel weighted MISO channels is first derived. Finally, we deduce the DMT of a broadcast channel using DPC and ZF precoders.

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arXiv2008

Distributed Space-Time Block Codes for the MIMO Multiple Access Channel

In this work, the Multiple transmit antennas Multiple Access Channel is considered. A construction of a family of distributed space-time codes for this channel is proposed. No Channel Side Information at the transmitters is assumed and users are not allowed to cooperate together. It is shown that the proposed code achieves the Diversity Multiplexing Tradeoff of the channel. As an example, we consider the two-user MIMO-MAC channel. Simulation results show the significant gain offered by the new coding scheme compared to an orthogonal transmission scheme, e.g. time sharing.

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arXiv2008

Distributed Space Time Codes for the Amplify-and-Forward Multiple-Access Relay Channel

In this work, we present a construction of a family of space-time block codes for a Multi-Access Amplify-and- Forward Relay channel with two users and a single half-duplex relay. It is assumed that there is no Channel Side Information at the transmitters and that they are not allowed to cooperate together. Using the Diversity Multiplexing Tradeoff as a tool to evaluate the performance, we prove that the proposed scheme is optimal in some sense. Moreover, we provide numerical results which show that the new scheme outperforms the orthogonal transmission scheme, e. g. time sharing and offers a significant gain.

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