arXiv subjects

Joseph J. Boutros

Publications and source records attributed to Joseph J. Boutros.

At least 19 recordsLinked to original sources

arXiv2026

Diversity in Coded TE-QKD Channels: Achieving Infinite Diversity out of Finite System Resources

We establish conditions and give proofs on how an error-correcting code can attain infinite diversity in a time-entanglement quantum key distribution (TE-QKD) reconciliation. The shocking result, never encountered in the literature on coding and communication theory, is that a decoder exhibits an infinite diversity order while the channel has finite diversity and the code has a relatively short finite length. This paper studies the diversity order of coded TE-QKD reconciliation, defined by the asymptotic slope of the error probability at high signal-to-noise ratio. For bounded-distance algebraic decoding, we derive a necessary and sufficient condition in terms of the number of photons per codeword and the decoding radius. For soft-decision decoding, we introduce the maximal finite diversity (MFD) property and prove that infinite diversity is achieved if and only if the code is MFD deficient. The infinite diversity in TE-QKD has no counterpart in classical fading channels, where decoding can only multiply a finite diversity order by a finite factor. Examples of short codes based on Golay, Reed-Solomon, Bose-Chaudhuri-Hocquenghem (BCH), and Reed-Muller codes validate the analysis and illustrate how the TE-QKD system parameters and the relatively short code parameters affect the achievable diversity for both hard and soft information reconciliation.

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arXiv2023

Time-Entanglement QKD: Secret Key Rates and Information Reconciliation Coding

In time entanglement-based quantum key distribution (QKD), Alice and Bob extract the raw key bits from the (identical) arrival times of entangled photon pairs by time-binning. Each of them individually discretizes time into bins and groups them into frames. They retain only the frames with a single occupied bin. Thus, Alice and Bob can use the position of the occupied bin within a frame to generate random key bits, as in PPM modulation. Because of entanglement, their occupied bins and their keys should be identical. However, practical photon detectors suffer from time jitter errors. These errors cause discrepancies between Alice's and Bob's keys. Alice sends information to Bob through the public channel to reconcile the keys. The amount of information determines the secret key rate. This paper computes the secret key rates possible with detector jitter errors and constructs codes for information reconciliation to approach these rates.

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arXiv2021

On the decoding of lattices constructed via a single parity check

This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice is constructed as a simple application of single parity-check principle on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check or via the k-ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided.

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arXiv2021

Neural network approaches to point lattice decoding

We characterize the complexity of the lattice decoding problem from a neural network perspective. The notion of Voronoi-reduced basis is introduced to restrict the space of solutions to a binary set. On the one hand, this problem is shown to be equivalent to computing a continuous piecewise linear (CPWL) function restricted to the fundamental parallelotope. On the other hand, it is known that any function computed by a ReLU feed-forward neural network is CPWL. As a result, we count the number of affine pieces in the CPWL decoding function to characterize the complexity of the decoding problem. It is exponential in the space dimension $n$, which induces shallow neural networks of exponential size. For structured lattices we show that folding, a technique equivalent to using a deep neural network, enables to reduce this complexity from exponential in $n$ to polynomial in $n$. Regarding unstructured MIMO lattices, in contrary to dense lattices many pieces in the CPWL decoding function can be neglected for quasi-optimal decoding on the Gaussian channel. This makes the decoding problem easier and it explains why shallow neural networks of reasonable size are more efficient with this category of lattices (in low to moderate dimensions).

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arXiv2020

On the decoding of Barnes-Wall lattices

We present new efficient recursive decoders for the Barnes-Wall lattices based on their squaring construction. The analysis of the new decoders reveals a quasi-quadratic complexity in the lattice dimension and a quasi-linear complexity in the list-size. The error rate is shown to be close to the universal lower bound in dimensions 64 and 128.

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arXiv2020

A lattice-based approach to the expressivity of deep ReLU neural networks

We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in $n$. We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU networks with quadratic depth and linear width in the space dimension. We also investigate the approximation error of one of these functions by shallower networks and prove a separation result. The main difference between our functions and other constructions is their practical interest: they arise in the scope of channel coding. Hence, computing such functions amounts to performing a decoding operation.

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arXiv2019

Entropy Estimation of Physically Unclonable Functions via Chow Parameters

A physically unclonable function (PUF) is an electronic circuit that produces an intrinsic identifier in response to a challenge. These identifiers depend on uncontrollable variations of the manufacturing process, which make them hard to predict or to replicate. Various security protocols leverage on such intrinsic randomness for authentification, cryptographic key generation, anti-counterfeiting, etc. Evaluating the entropy of PUFs (for all possible challenges) allows one to assess the security properties of such protocols. In this paper, we estimate the probability distribution of certain kinds of PUFs composed of $n$ delay elements. This is used to evaluate relevant Rényi entropies and determine how they increase with $n$. Such a problem was known to have extremely high complexity (in the order of $2^{2^n}$) and previous entropy estimations were carried out up to $n=7$. Making the link with the theory of Boolean threshold functions, we leverage on the representation by Chow parameters to estimate probability distributions up to $n=10$. The resulting Shannon entropy of the PUF is close to the max-entropy, which is asymptotically quadratic in $n$.

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arXiv2019

On the CVP for the root lattices via folding with deep ReLU neural networks

Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in Rn, known as the closest vector problem (CVP), becomes a classification problem in the fundamental parallelotope with a piecewise linear function defining the boundary. Theoretical results are obtained by studying root lattices. We show how the number of pieces in the boundary function reduces dramatically with folding, from exponential to linear. This translates into a two-layer ReLU network requiring a number of neurons growing exponentially in n to solve the CVP, whereas this complexity becomes polynomial in n for a deep ReLU network.

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arXiv2019

Multilevel MIMO Detection with Deep Learning

A quasi-static flat multiple-antenna channel is considered. We show how real multilevel modulation symbols can be detected via deep neural networks. A multi-plateau sigmoid function is introduced. Then, after showing the DNN architecture for detection, we propose a twin-network neural structure. Batch size and training statistics for efficient learning are investigated. Near-Maximum-Likelihood performance with a relatively reasonable number of parameters is achieved.

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arXiv2019

Neural Lattice Decoders

Lattice decoders constructed with neural networks are presented. Firstly, we show how the fundamental parallelotope is used as a compact set for the approximation by a neural lattice decoder. Secondly, we introduce the notion of Voronoi-reduced lattice basis. As a consequence, a first optimal neural lattice decoder is built from Boolean equations and the facets of the Voronoi cell. This decoder needs no learning. Finally, we present two neural decoders with learning. It is shown that L1 regularization and {\em a priori} information about the lattice structure lead to a simplification of the model.

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arXiv2018

Exploring and Experimenting with Shaping Designs for Next-Generation Optical Communications

A class of circular 64-QAM that combines 'geometric' and 'probabilistic' shaping aspects is presented. It is compared to square 64-QAM in back-to-back, single-channel, and WDM transmission experiments. First, for the linear AWGN channel model, it permits to operate close to the Shannon limits for a wide range of signal-to-noise ratios. Second, WDM simulations over several hundreds of kilometers show that the obtained signal-to-noise ratios are equivalent to - or slightly exceed - those of probabilistic shaped 64-QAM. Third, for real-life validation purpose, an experimental comparison with unshaped 64-QAM is performed where 28% distance gains are recorded when using 19 channels at 54.2 GBd. This again is in line - or slightly exceeds - the gains generally obtained with probabilistic shaping. Depending upon implementation requirements (core forward-error correcting scheme for example), the investigated modulation schemes may be key alternatives for next-generation optical systems.

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arXiv2017

Leech Constellations of Construction-A Lattices

The problem of communicating over the additive white Gaussian noise (AWGN) channel with lattice codes is addressed in this paper. Theoretically, Voronoi constellations have proved to yield very powerful lattice codes when the fine/coding lattice is AWGN-good and the coarse/shaping lattice has an optimal shaping gain. However, achieving Shannon capacity with these premises and practically implementable encoding algorithms is in general not an easy task. In this work, a new way to encode and demap Construction-A Voronoi lattice codes is presented. As a meaningful application of this scheme, the second part of the paper is focused on Leech constellations of low-density Construction-A (LDA) lattices: LDA Voronoi lattice codes are presented whose numerically measured waterfall region is situated at less than 0.8 dB from Shannon capacity. These LDA lattice codes are based on dual-diagonal nonbinary low-density parity-check codes. With this choice, encoding, iterative decoding, and demapping have all linear complexity in the blocklength.

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arXiv2017

Probabilistic Shaping and Non-Binary Codes

We generalize probabilistic amplitude shaping (PAS) with binary codes to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping.

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arXiv2016

LDA Lattices Without Dithering Achieve Capacity on the Gaussian Channel

This paper deals with Low-Density Construction-A (LDA) lattices, which are obtained via Construction A from non-binary Low-Density Parity-Check codes. More precisely, a proof is provided that Voronoi constellations of LDA lattices achieve the capacity of the AWGN channel under lattice encoding and decoding. This is obtained after showing the same result for more general Construction-A lattice constellations. The theoretical analysis is carried out in a way that allows to describe how the prime number underlying Construction A behaves as a function of the lattice dimension. Moreover, no dithering is required in the transmission scheme, simplifying some previous solutions of the problem. Remarkably, capacity is achievable with LDA lattice codes whose parity-check matrices have constant row and column Hamming weights. Some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound of the minimum Euclidean distance of LDA lattices in our ensemble.

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arXiv2016

Edge Coloring and Stopping Sets Analysis in Product Codes with MDS components

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$. Stopping sets are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d_1+1)(d_2+1)$, where $d_1$ and $d_2$ are the minimum Hamming distances of the column and row MDS components respectively. Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph})$, for a given differential evolution parameter $\aleph$, where $M^{\aleph}$ itself is small with respect to the huge cardinality of the coloring ensemble. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both block and independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.

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arXiv2014

Lattices over Eisenstein Integers for Compute-and-Forward

In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of nested lattice codebooks over Eisenstein integers surpasses lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity.

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arXiv2012

Time-Varying Space-Only Codes for Coded MIMO

Multiple antenna (MIMO) devices are widely used to increase reliability and information bit rate. Optimal error rate performance (full diversity and large coding gain), for unknown channel state information at the transmitter and for maximal rate, can be achieved by approximately universal space-time codes, but comes at a price of large detection complexity, infeasible for most practical systems. We propose a new coded modulation paradigm: error-correction outer code with space-only but time-varying precoder (as inner code). We refer to the latter as Ergodic Mutual Information (EMI) code. The EMI code achieves the maximal multiplexing gain and full diversity is proved in terms of the outage probability. Contrary to most of the literature, our work is not based on the elegant but difficult classical algebraic MIMO theory. Instead, the relation between MIMO and parallel channels is exploited. The theoretical proof of full diversity is corroborated by means of numerical simulations for many MIMO scenarios, in terms of outage probability and word error rate of LDPC coded systems. The full-diversity and full-rate at low detection complexity comes at a price of a small coding gain loss for outer coding rates close to one, but this loss vanishes with decreasing coding rate.

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arXiv2011

Low-Density Graph Codes for slow fading Relay Channels

We study Low-Density Parity-Check (LDPC) codes with iterative decoding on block-fading (BF) Relay Channels. We consider two users that employ coded cooperation, a variant of decode-and-forward with a smaller outage probability than the latter. An outage probability analysis for discrete constellations shows that full diversity can be achieved only when the coding rate does not exceed a maximum value that depends on the level of cooperation. We derive a new code structure by extending the previously published full-diversity root-LDPC code, designed for the BF point-to-point channel, to exhibit a rate-compatibility property which is necessary for coded cooperation. We estimate the asymptotic performance through a new density evolution analysis and the word error rate performance is determined for finite length codes. We show that our code construction exhibits near-outage limit performance for all block lengths and for a range of coding rates up to 0.5, which is the highest possible coding rate for two cooperating users.

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