Yair Be'ery

Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes

Maximum-likelihood soft-decision decoding of linear block codes is addressed. A binary multiple-check generalization of the Wagner rule is presented, and two methods for its implementation, one of which resembles the suboptimal Forney-Chase algorithms, are described. Besides efficient soft decoding of small codes, the generalized rule enables utilization of subspaces of a wide variety, thereby yielding maximum-likelihood decoders with substantially reduced computational complexity for some larger binary codes. More sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the (24, 12) Golay code, yielding a computational gain factor of about 2 with respect to previous methods. A ternary single-check version of the Wagner rule is applied for efficient soft decoding of the (12, 6) ternary Golay code. >

Optimal soft decision block decoders based on fast Hadamard transform

An approach for efficient utilization of fast Hadamard transform in decoding binary linear block codes is presented. Computational gain is obtained by employing various types of concurring codewords, and memory reduction is also achieved by appropriately selecting rows for the generator matrix. The availability of these codewords in general, and particularly in some of the most frequently encountered codes, is discussed.

The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions

Multilevel constructions of the binary Golay code and the Leech lattice are described. Both constructions are based upon the projection of the Golay code and the Leech lattice onto the (6,3,4) hexacode over GF(4). However, unlike the previously reported constructions, the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates. Based upon the multilevel structure of the Golay code and the Leech lattice, efficient bounded-distance decoding algorithms are devised. The bounded-distance decoder for the binary Golay code requires at most 431 operations. As compared to 651 operations for the best known maximum-likelihood decoder. Efficient bounded-distance decoding of the Leech lattice is achieved by means of partitioning it into four cosets of Q/sub 24/, beyond the conventional partition into two H/sub 24/ cosets. The complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case, as compared to about 3600 operations for the best known in maximum-likelihood decoder. It is shown that the proposed algorithms decode correctly at least up to the guaranteed error-correction radius of the maximum-likelihood decoder. Thus, the loss in coding-gain is due primarily to an increase in the effective error-coefficient, which is calculated exactly for both algorithms. Furthermore, the performance of the Leech lattice decoder on the AWGN channel is evaluated experimentally by means of a comprehensive computer simulation. The results show a loss in coding-gain of less than 0.1 dB relative to the maximum-likelihood decoder for BER ranging from 10/sup -1/ to 10/sup -7/. >

Bounds on the trellis size of linear block codes

The size of minimal trellis representation of linear block codes is addressed. Two general upper bounds on the trellis size, based on the zero-concurring codewords and the contraction index of the subcodes, are presented. The related permutations for attaining the bounds are exhibited. These bounds evidently improve the previously published general bound. Additional bounds based on certain code constructions are derived. The focus is on the squaring construction, and specific constructive bounds for Reed-Muller and repeated-root cyclic codes are obtained. In particular, the recursive squaring construction of Reed-Muller codes is explored and the exact minimal trellis size of this design is obtained. Efficient permutations, in the sense of the trellis size, are also demonstrated by using shortening and puncturing methods. The corresponding bounds are specified. >

Maximum likelihood decoding of the Leech lattice

An algorithm for maximum likelihood decoding of the Leech lattice is presented. The algorithm involves projecting the points of the Leech lattice directly onto the codewords of the (6,3,4) quaternary code-the hexacode. Projection on the hexacode induces a partition of the Leech lattice into four cosets of a certain sublattice 24. Such a partition into cosets enables maximum likelihood decoding of the Leech lattice with 3595 real operations in the worst case and only 2955 operations on the average. This is about half the worst case and the average complexity of the best previously known algorithm. >

Fast decoding of the Leech lattice

An efficient algorithm is presented for maximum-likelihood soft-decision decoding of the Leech lattice. The superiority of this decoder with respect to both computational and memory complexities is demonstrated in comparison with previously published decoding methods. Gain factors in the range of 2-10 are achieved. The authors conclude with some more advanced ideas for achieving a further reduction of the algorithm complexity based on a generalization of the Wagner decoding method to two parity constraints. A comparison with the complexity of some trellis-coded modulation schemes is discussed. The decoding algorithm presented seems to achieve a computational complexity comparable to that of the equivalent trellis codes. >

Convergence analysis of turbo-decoding of product codes

Geometric interpretation of turbo decoding has founded an analytical basis, and provided tools for the analysis of this algorithm. We focus on turbo decoding of product codes, and based on the geometric framework, we extend the analytical results and show how analysis tools can be practically adapted for this case. Specifically, we investigate the algorithms stability and its convergence rate. We present new results concerning the structure and properties of stability matrices of the algorithm, and develop upper bounds on the algorithms convergence rate. We prove that for any 2/spl times/2 (information bits) product codes, there is a unique and stable fixed point. For the general case, we present sufficient conditions for stability. The interpretation of these conditions provides an insight to the behavior of the decoding algorithm. Simulation results, which support and extend the theoretical analysis, are presented for Hamming [(7,4,3)]/sup 2/ and Golay [(24,12,8)]/sup 2/ product codes.

Learning to decode linear codes using deep learning

A novel deep learning method for improving the belief propagation algorithm is proposed. The method generalizes the standard belief propagation algorithm by assigning weights to the edges of the Tanner graph. These edges are then trained using deep learning techniques. A well-known property of the belief propagation algorithm is the independence of the performance on the transmitted codeword. A crucial property of our new method is that our decoder preserved this property. Furthermore, this property allows us to learn only a single codeword instead of exponential number of codewords. Improvements over the belief propagation algorithm are demonstrated for various high density parity check codes.

Delayed adaptive LMS filtering: current results

The main results are presented of an analysis of the convergence and the steady-state behavior of the DLMS (delayed least mean square) algorithm, with the aim of providing useful insight which may be helpful in the design of such filters. The problem is defined, and some basic definitions are presented. Conditions for convergence, convergence rate, and limits are discussed. Remarks are presented. The implications of the results for the design of DLMS adaptive filters are addressed.<<ETX>>

Soft trellis-based decoder for linear block codes

A systematic design of a trellis-based maximum-likelihood soft-decision decoder for linear block codes is presented. The essence of the decoder is to apply an efficient search algorithm for the error pattern on a reduced trellis representation of a certain coset. Rather than other efficient decoding algorithms, the proposed decoder is systematically designed for long codes, as well as for short codes. Computational gain of up to 6 is achieved for long high-rate codes over the well-known trellis decoder of Wolf (1978). Efficient decoders are also obtained for short and moderate length codes. >