Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson , Kerdock, Preparata, Goethals, and Delsarte-Goethals . It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z 4 , the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z 4 domain implies that the binary images have dual weight distributions. The Kerdock and 'Preparata' codes are duals over Z 4 -and the Nordstrom-Robinson code is self-dual -which explains why their weight distributions are dual to each other. The Kerdock and 'Preparata' codes are Z 4 -analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z 4 , which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the 'Preparata' code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first-and second-order Reed-Muller codes are also linear over Z 4 , but extended Hamming codes of length n ≥ 32 and the Golay code are not. Using Z 4 -linearity, a new family of distance regular graphs are constructed on the cosets of the 'Preparata' code.The version of the Kerdock code that we use is the standard one, while our version of the Preparata code differs from the standard one in that it is not a subcode of the extended Hamming code but of a nonlinear code with the same weight distribution as the extended Hamming code. Our 'Preparata' code has the same weight distribution as Preparata's version, and has a similar construction in terms of finite field transforms. In our version, the Kerdock and 'Preparata' codes are Z 4 -analogues of first-order Reed-Muller and extended Hamming codes, respectively. Since the new construction is so simple, we propose that this is the 'correct' way to define these codes.The situation may be compared with that for Hamming codes. It is known that there are many binary codes with the same weight distribution as the Hamming code -all are perfect single-error correcting codes, but one is distinguished by being linear (see [73], [59], [60] and also §5.4). Similarly, there are many binary codes with the same weight distributions as the Kerdock and Preparata codes; one pair is distinguished by being the images of a dual pair of linear extended-cyclic codes over Z 4 . It happens that Kerdock picked out the distinguished code, although Preparata did not. Kerdock and Preparata codes exist for all lengths n = 4 m ≥ 16. At length 16 they coincide, giving the Nordstrom-Robinson code [58], [66], [33]. The Z 4 version of the Nordstrom-Robinson code turns out to be the 'octacode' [22], [23], a self-dual code of length 8 over Z 4 that is used when the Leech lattice is constructed from eight copies of the face-centered cubic lattice. The very goo...

The design of space-time codes to achieve full spatial diversity over fading channels has largely been addressed by handcrafting example codes using computer search methods and only for small numbers of antennas. The lack of more general designs is in part due to the fact that the diversity advantage of a code is the minimum rank among the complex baseband differences between modulated codewords, which is difficult to relate to traditional code designs over finite fields and rings. In this paper, we present general binary design criteria for PSK-modulated space-time codes. For linear BPSK/QPSK codes, the rank of (binary projections of) the unmodulated codewords, as binary matrices over the binary field, is a sufficient design criterion: full binary rank guarantees full spatial diversity. This criterion accounts for much of what is currently known about PSK-modulated space-time codes. We develop new fundamental code constructions for both quasi-static and time-varying channels. These are perhaps the first general constructions-other than delay diversity schemes-that guarantee full spatial diversity for an arbitrary number of transmit antennas.

In this paper, we introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece. Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. Index Terms-Convergence-based design, graphical codes, iterative decoding, low-density parity-check (LDPC) codes, turbo codes. I. INTRODUCTION O NE of the main reasons behind the impressive performance achieved by graphical codes such as parallel concatenated convolutional code(s) (PCCC) is the elegant iterative decoding algorithm with the exchange of soft information between successive iterations. Recently, it has been shown that this iterative decoding algorithm is an instance of Pearl belief propagation in Bayesian networks [1]. Whereas belief propagation is known to converge to the maximum a posteriori (MAP) solution for graphs without loops, relatively little progress has been achieved to date in understanding the theoretical behavior of the algorithm on graphs with loops, especially as a suboptimal decoder for codes having such graphical representation [1]. The rediscovery of Gallager's low-density parity-check (LDPC) codes [2] and Wiberg's work on the graphical repre

The information-theoretic capacity of multiple antenna systems was shown to be significantly higher than that of single antenna systems in Rayleigh-fading channels. In an attempt to realize this capacity, Foschini proposed the layered space-time architecture. This scheme was argued to asymptotically achieve a lower bound on the capacity. Another line of work has focused on the design of channel codes that exploit the spatial diversity provided by multiple transmit antennas [2], [3]. In this paper, we take a fresh look at the problem of designing multiple-input-multiple-output (MIMO) wireless systems. First, we develop a generalized framework for the design of layered space-time systems. Then, we present a novel layered architecture that combines efficient algebraic code design with iterative signal processing techniques. This novel layered system is referred to as the threaded space-time (TST) architecture. The TST architecture provides more flexibility in the tradeoff between power efficiency, bandwidth efficiency, and receiver complexity. It also allows for exploiting the temporal diversity provided by time-varying fading channels. Simulation results are provided for the various techniques that demonstrate the superiority of the proposed TST architecture over both the diagonal layered space-time architecture in [1] and the recently proposed multilayering approach [4].

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