Erik Agrell

Closest point search in lattices

In this semitutorial paper, a comprehensive survey of closest point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest point search algorithm, based on the Schnorr-Euchner (1995) variation of the Pohst (1981) method, is implemented. Given an arbitrary point x /spl isin/ /spl Ropf//sup m/ and a generator matrix for a lattice /spl Lambda/, the algorithm computes the point of /spl Lambda/ that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan (1983, 1987) algorithm and an experimental comparison with the Pohst (1981) algorithm and its variants, such as the Viterbo-Boutros (see ibid. vol.45, p.1639-42, 1999) decoder. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, computing the Voronoi (1908)-relevant vectors, and finding a Korkine-Zolotareff (1873) reduced basis.

Power-Efficient Modulation Formats in Coherent Transmission Systems

Coherent optical transmission systems have a four-dimensional (4-D) signal space (two quadratures in two polarizations). These four dimensions can be used to create modulation formats that have a better power efficiency (higher sensitivity) than the conventional binary phase shift keying/quadrature phase shift keying (BPSK/QPSK) signals. Several examples are given, with some emphasis on a 24-level format and an 8-level format, including descriptions of how they can be realized and expressions for their symbol and bit error probabilities. These formats are, respectively, an extension and a subset of the commonly used 16-level dual-polarization QPSK format. Sphere packing simulations in 2, 3, and 4 dimensions, up to 32 levels, are used to verify their optimality. The numerical results, as the number of levels increases, are shown to agree with lattice-theoretical results. Finally, we point out that the use of these constellations will lead to improved fundamental sensitivity limits for optical communication systems, and they may also be relevant as a way of reducing power demands and/or nonlinear influence.

Which is the most power-efficient modulation format in optical links?

By exploiting the electromagnetic waves four-dimensional signal space, we find that for the additive white Gaussian noise channel, the modulation format with best sensitivity to be an 8-level format with 1.76 dB asymptotic gain over BPSK, for uncoded optical transmission with coherent detection. Low-complexity modulators are presented for the format, as well as an interpretation in terms of quantum-limited sensitivity.

Upper bounds for constant-weight codes

Let A(n,d,w) denote the maximum possible number of codewords in an (n,d,w) constant-weight binary code. We improve upon the best known upper bounds on A(n,d,w) in numerous instances for n/spl les/24 and d/spl les/12, which is the parameter range of existing tables. Most improvements occur for d=8, 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n/spl les/28 and d/spl les/14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n,d,w) by means of mapping constant-weight codes into Euclidean space. This approach produces, among other results, a bound on A(n,d,w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-bounded-weight codes, which may be thought of as a generalization of the doubly-constant-weight codes. Subsequently, a class of Euclidean-space codes, called zonal codes, is introduced, and a bound on the size of such codes is established. This is used to derive bounds for doubly-bounded-weight codes, which are in turn used to derive bounds on A(n,d,w). We also develop a universal method to establish constraints that augment the Delsarte inequalities for constant-weight codes, used in the linear programming bound. In addition, we present a detailed survey of known upper bounds for constant-weight codes, and sharpen these bounds in several cases. All these bounds, along with all known dependencies among them, are then combined in a coherent framework that is amenable to analysis by computer. This improves the bounds on A(n,d,w) even further for a large number of instances of n, d, and w.

On the optimality of the binary reflected Gray code

This paper concerns the problem of selecting a binary labeling for the signal constellation in M-PSK, M-PAM, and M-QAM communication systems. Gray labelings are discussed and the original work by Frank Gray is analyzed. As is noted, the number of distinct Gray labelings that result in different bit-error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings, will give the lowest possible average probability of bit errors for the considered constellations. Under certain assumptions on the channel, the answer is that the labeling proposed by Gray, the binary reflected Gray code, is the optimal labeling for all three constellations, which has, surprisingly, never been proved before.

The Hadamard transform-a tool for index assignment

We show that the channel distortion for maximum-entropy encoders, due to noise on a binary-symmetric channel, is minimized if the vector quantizer can be expressed as a linear transform of a hypercube. The index assignment problem is regarded as a problem of linearizing the vector quantizer. We define classes of index assignments with related properties, within which the best index assignment is found by sorting, not searching. Two powerful algorithms for assigning indices to the codevectors of nonredundant coding systems are presented. One algorithm finds the optimal solution in terms of linearity, whereas the other finds a very good, but suboptimal, solution in a very short time.

Computation of the exact bit-error rate of coherent M-ary PSK with Gray code bit mapping

The problem of calculating the average bit-error probability (BEP) of coherent M-ary phase-shift keying (PSK) over a Gaussian channel has been studied previously in the literature. A solution to the problem for systems using a binary reflected Gray code (BRGC) to map bits to symbols was first presented by P.J. Lee (see ibid., vol.COM-34, p.488-91, 1986). We show that the results obtained by Lee are incorrect for M/spl ges/16. We show that the reason for this is an invalid assumption that the bit-error rate (BER) is independent of the transmitted symbols, an assumption which has also propagated to textbooks. We give a new expression for the BER of M-PSK systems using the BRGC and compare this with Lees results.

Roadmap of optical communications

Quantum physics allows one to produce truly random bits. Moreover, it allows one to distribute them in such a way that one can certify their privacy before eventually using them for cryptography applications. Quantum Random Number generators (QRNG) and Quantum Key Distribution (QKD) have found a few niche markets. Today, some commercial clients use QKD continuously 24×7 a week. In this workshop world specialists will talk about reliability tests in quantum networks; about quantum hacking, its importance and limitations, and its role in classical and quantum cryptography; about high rate and about low cost QKD systems; about free space quantum communication; and about future quantum repeaters for continental scale quantum communication.Lightwave communications is a necessity for the information age. Optical links provide enormous bandwidth, and the optical fiber is the only medium that can meet the modern societys needs for transporting massive amounts of data over long distances. Applications range from global high-capacity networks, which constitute the backbone of the internet, to the massively parallel interconnects that provide data connectivity inside datacenters and supercomputers. Optical communications is a diverse and rapidly changing field, where experts in photonics, communications, electronics, and signal processing work side by side to meet the ever-increasing demands for higher capacity, lower cost, and lower energy consumption, while adapting the system design to novel services and technologies. Due to the interdisciplinary nature of this rich research field, Journal of Optics has invited 16 researchers, each a world-leading expert in their respective subfields, to contribute a section to this invited review article, summarizing their views on state-of-the-art and future developments in optical communications.

4-PAM for high-speed short-range optical communications

In this work, we compare 4-pulse amplitude modulation and on-off keying modulation formats at high speed for short-range optical communication systems. The transmission system comprised a directly modulated vertical-cavity surface-emitting laser operating at a wavelength of 850 nm, an OM3+ multimode fiber link, and a photodetector detecting the intensity at the receiver end. The modulation formats were compared both at the same bit-rate and at the same symbol rate. The maximum bit-rate used was 25 Gbps. Propagation distances up to 600 m were investigated at 12.5 Gbps. All measurements were done in real time and without any equalization.

Frequency diversity performance of coded multiband-OFDM systems on IEEE UWB channels

This paper investigates how convolutional and Reed-Solomon codes can be used to improve the performance of multiband-OFDM by utilizing the inherent frequency diversity of the new IEEE 802.15 UWB channel models. A normalized amplitude autocovariance function of the Fourier transform of the channel impulse response is defined. Then the average coherence bandwidths of CM1, CM2, CM3, and CM4 are estimated to be 31.6, 16.3, 11.0, 5.8 MHz, respectively. Using the central limit theorem, we can expect that the performance of an uncoded OFDM system on CM1-CM4 without shadowing is the same as in a Rayleigh fading channel with uniformly distributed phase. The performance of a convolutional code with rate 1/2 and constraint length 7 on CM2-CM4 without shadowing are up to 0.4 dB worse than that of on an uncorrelated Rayleigh fading channel. The loss for CM1 is around 1 dB. A block interleaver with 32 rows and 24 columns was used. This result is also valid for a convolutional code with rate 1/4 and constraint length 7. For code rates around 2/3, the performance of a punctured convolutional code with soft-decision decoding is much better than that of the Reed-Solomon codes with with 6, 7, and 8 bits per symbol and hard-decision decoding.