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  • Agrell, Erik, 1965-, et al. (författare)
  • A table of upper bounds for binary codes
  • 2001
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 47:7, s. 3004-3006
  • Tidskriftsartikel (refereegranskat)abstract
    • Let A(n, d) denote the maximum possible number of codewords in an (n, d) binary code. We establish four new bounds on A(n, d), namely, A(21, 4)⩽43689, A(22, 4)⩽87378, A(22, 6)⩽6941, and A(23, 4)⩽173491. Furthermore, using previous upper bounds on the size of constant-weight binary codes, we reapply known methods to generate a table of bounds on A(n, d) for all n⩽28. This table extends the range of parameters compared with previously known tables.
  • Agrell, Erik, 1965-, et al. (författare)
  • Closest point search in lattices
  • 2002
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 48:8, s. 2201-2214
  • Tidskriftsartikel (refereegranskat)abstract
    • 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 variation of the Pohst method, is implemented. Given an arbitrary point x is an element of R-m and a generator matrix for a lattice A, the algorithm computes the point of A 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 algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent Viterbo-Boutros 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-relevant vectors, and finding. a Korkine-Zolotareff reduced basis.
  • Agrell, Erik, 1965-, et al. (författare)
  • On the optimality of the binary reflected Gray code
  • 2004
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 50:12, s. 3170-3182
  • Tidskriftsartikel (refereegranskat)abstract
    • 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.
  • Agrell, Erik, 1965- (författare)
  • On the Voronoi neighbor ratio for binary linear block codes
  • 1998
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 44:7, s. 3064-3072
  • Tidskriftsartikel (refereegranskat)abstract
    • Soft-decision decoding of block codes is regarded as the geometrical problem of identifying the Voronoi region within which a given input vector lies. A measure, called the neighbor ratio, is proposed to characterize how many facets a Voronoi region has. Theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes. An asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1/2 or not. For rates below this threshold, all pairs of codewords tend to share a Voronoi facet; for higher rates, a relatively small fraction of them do.
  • Agrell, Erik, 1965-, et al. (författare)
  • Optimal Alphabets and Binary Labelings for BICM at Low SNR
  • 2011
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 57:10, s. 6650-6672
  • Tidskriftsartikel (refereegranskat)abstract
    • Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e. constellations that make BICM achieve the Shannon limit $-1.59 \tr{dB}$. It is shown that the $\Eb/N_0$ required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.
  • Agrell, Erik, 1965-, et al. (författare)
  • Optimization of lattices for quantization
  • 1998
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 44:5, s. 1814-1828
  • Tidskriftsartikel (refereegranskat)abstract
    • A training algorithm for the design of lattices for vector quantization is presented. The algorithm uses a steepest descent method to adjust a generator matrix, in the search for a lattice whose Voronoi regions have minimal normalized second moment. The numerical elements of the found generator matrices are interpreted and translated into exact values. Experiments show that the algorithm is stable, in the sense that several independent runs reach equivalent lattices. The obtained lattices reach as low second moments as the best previously reported lattices, or even lower. Specifically, we report lattices in nine and ten dimensions with normalized second moments of 0.0716 and 0.0708, respectively, and nonlattice tessellations in seven and nine dimensions with 0.0727 and 0.0711, which improves on previously known values. The new nine- and ten-dimensional lattices suggest that Conway and Sloane's (1993) conjecture on the duality between the optimal lattices for packing and quantization might be false. A discussion of the application of lattices in vector quantizer design for various sources, uniform and nonuniform, is included.
  • Agrell, Erik, 1965-, et al. (författare)
  • Signal Shaping for BICM at Low SNR
  • 2013
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 59:4, s. 2396-2410
  • Tidskriftsartikel (refereegranskat)abstract
    • The generalized mutual information (GMI) of bit-interleaved coded modulation (BICM) systems, sometimes called the BICM capacity, is investigated at low signal-to-noise ratio (SNR). The combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit are completely characterized. The main conclusion is that a BICM system with probabilistic shaping achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR BICM-GMI, in addition to what is provided by geometrical shaping. The analytical conclusions are confirmed by numerical results, which also show that for a fixed input alphabet, probabilistic shaping can improve the BICM-GMI in the low and medium SNR range.
  • Agrell, Erik, 1965-, et al. (författare)
  • Upper bounds for constant-weight codes
  • 2000
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 46:7, s. 2373-2395
  • Tidskriftsartikel (refereegranskat)abstract
    • 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⩽24 and d⩽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⩽28 and d⩽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.
  • Agrell, Erik, 1965- (författare)
  • Voronoi regions for binary linear block codes
  • 1996
  • Ingår i: IEEE Transactions on Information Theory. - 0018-9448. ; 42:1, s. 310-316
  • Tidskriftsartikel (refereegranskat)abstract
    • The Voronoi regions of a block code govern many aspects of the code's performance on a Gaussian channel, and they are fundamental instruments in, for example, error probability analysis and soft-decision decoding. The article presents an efficient method for finding the boundaries of the Voronoi regions for an arbitrary binary linear block code. Two theoretical results together lead to the Voronoi regions. First, it is shown that the question of the Voronoi neighborship can be reduced into testing a simpler relation, called the Gabriel neighborship. Second, a fast method of recognising Gabriel neighbors is proposed. These results are finally employed to describe the Voronoi regions for the Golay codes and several BCH codes, including Hamming codes.
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