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Maximum likelihood sequence estimation (MLSE) in the shape of the Viterbi algorithm, although a strictly hard-output approach, is utilized for iterative decoding of turbo codes. It is a practical alternative when decoding complexity and speed may be traded for performance. Index Terms--Turbo code, iterative decoding.

The standard product construction is discussed with respect to nonlinear codes. Thus, so-called nonlinearproduct codes are obtained that are better than linear product codes of similar length and code rate, and at the same time, amenable for encoding/decoding. On the other hand, it is shown that certain notorious nonlinear codes have an augmented product construction, namely, they can be constructed by taking the union of a product code and certain of its cosets. The binary Hamming codes are shown to have similar construction. A simple two-stage decoder is proposed for nonlinear product (NLP) codes. The decoder is shown to be a bounded-distance (BD) information decoder that is the nonlinear equivalent of the BD decoder employed for linear codes. A list-based maximum-likelihood decoder is also discussed. Index Terms--Bounded-distance decoding, nonlinear codes, product codes.

This letter investigates the so-called product lattice (PL) construction. A PL is obtained from two low-dimensional lattices by means of the Kronecker product. Fundamental properties, mainly those determining the performance of a lattice as an error-correcting scheme over a bandwidth-limited channel, are derived for PLs. Due to their special structure and properties, PLs can provide an attractive family of lattice codes of good performance/complexity tradeoffs. Index Terms--Barnes-Wall lattice, coset codes, lattice codes, Leech lattice, product codes.

Coded modulation scheme of high-coding gain is presented for flat fading channels. The scheme introduces a product construction that employs two types of codes: "standard" good codes and codes that are more suitable for the fading environment. Short-blocklength codes of unique features are, thus, obtained for the bandwidth-limited regime. Using this approach, codes of bandwidth efficiencies 1, 2, and 3 bits/s/Hz are detailed. An iterative decoding algorithm for this product construction is devised. It relies on separate columns/rows decoding linked by (reliability) information transfer. Simulation results reveal that these codes compare favorably with known codes of short block-length when perfect channel state information is available to the receiver. The proposed approach is easily extended for the frequency-selective fading channel with multicarrier signaling. Index Terms--Diversity transform, fading channels, iterative decoding/ estimation, product codes.

A binary product lattice is generated from two binary component lattices of lower dimensions by employing the Kronecker product. This work focuses on codes carved from binary product lattices. Defined as such, an intriguing problem is that of effectively mapping independent data sequences onto a selected subset of lattice points. A novel approach is disclosed yielding an explicit connection between source bits and lattice points. Decoding methods typically used for binary product codes do not apply for product lattices. Several alternative decoding approaches are discussed. In particular, a provably bounded-distance decoder is presented. It relies on the fact that a product lattice code point may be regarded as a two-dimensional array whose rows and columns are points in the component lattices. The obtained results are compared with classical lattices known in the art. Index Terms--Barnes-Wall lattice, binary code, binary lattice, bounded-distance decoding, lattice, maximum-likelihood (ML) decoding, nearest neighbors, product code, product lattice, Reed-Muller code.

This paper concerns the construction of the so-called augmented product codes and augmented product lattices. These are obtained by augmenting product codes or product lattices from certain classes thus obtaining higher dimensional codes or lattices from the same class, respectively. Certain properties of the augmented product construction are derived, and specific construction examples are given. In particular, it is shown that the Reed-Muller codes, the Golay code, the Barnes-Wall lattices, as well as the Leech lattice all have various augmented product constructions. Index Terms--Binary code, lattice, lattice/code (deep) holes, (principal sublattices of) Barnes-Wall lattice, product code, product lattice, (decomposable) binary lattice, Reed-Muller code.

We present a novel scheme that combines decision feedback equalization (DFE) with high-rate error-detection coding in an efficient manner. The proposed scheme is shown to considerably outperform the conventional practice on channels with high SNR, such as those encountered in twisted-pair telephonia systems. In order to analyze the performance of our method, we introduce an approximate mathematical model taking into account the error propagation phenomenon. Based on this model, upper and lower bounds on the overall probability of error are developed. These show that a simple low-redundancy error-detecting code, when properly integrated with the equalizer, can make the overall probability of error several orders of magnitude lower than that obtained with the conventional DFE, or with a DFE followed by an error-correcting code. Computer simulations of the proposed method have been performed for several channels, including the so-called high-bit-rate digital subscriber line (HDSL) test-loop #4, which is known to have a considerable amount of intersymbol interference. For all these channels, our results show that a reduction in the probability of error by more than three orders of magnitude can be obtained using codes of rate 0.96 and above. This, in turn, translates into power savings (coding gain) of 2.5 to 3 dB. Index Terms - Equalization, coding, decision-feedback, inter-symbol interference, trellis, HDSL.

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 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]. Index Terms - Leech lattice, Golay code, bounded-distance decoding, multilevel constructions.

A projection of binary Reed--Muller codes R(r, m) onto GF [(4).sup.m-2] is presented. For an R(r, m) code, this operation yields a linear quaternary code with the same length, dimension, and minimum distance as the Reed--Muller R(r - 1, m - 2) code. Based upon this projection, multilevel construction is given for R(r, m), where the constituent codes applied to the different levels are themselves the Reed--Muller codes R(r - 2, m - 2) and R(r, m - 2), as well as the aforementioned quaternary code. This construction of Reed--Muller codes is readily applicable for their efficient decoding. Index Terms--Mapping, multilevel, multistage, projection, Reed--Muller codes.