An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Contention-free interleavers have been recently shown to be suitable for parallel decoding of turbo codes. In this correspondence, it is shown that permutation polynomials generate maximum contention-free interleavers, i.e., every factor of the interleaver length becomes a possible degree of parallel processing of the decoder. Further, it is shown by computer simulations that turbo codes using these interleavers perform very well for the 3rd Generation Partnership Project (3GPP) standard.
It is well known that an interleaver with random properties, quite often generated by pseudo-random algorithms, is one of the essential building blocks of turbo codes. However, randomly generated interleavers have two major drawbacks: lack of an adequate analysis that guarantees their performance and lack of a compact representation that leads to a simple implementation. In this paper we present several new classes of deterministic interleavers of length , with construction complexity (), that permute a sequence of bits with nearly the same statistical distribution as a random interleaver and perform as well as or better than the average of a set of random interleavers. The new classes of deterministic interleavers have a very simple representation based on quadratic congruences and hence have a structure that allows the possibility of analysis as well as a straightforward implementation. Using the new interleavers, a turbo code of length 16384 that is only 0.7 dB away from capacy at a bit-error rate (BER) of 10 5 is constructed. We also generalize the theory of previously known deterministic interleavers that are based on block interleavers, and we apply this theory to the construction of a nonrandom turbo code of length 16384 with a very regular structure whose performance is only 1.1 dB away from capacity at a BER of 10 5 .
An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are important because they admit analytical designs and simple, practical hardware implementation. The spread factor of an interleaver is a common measure for turbo coding applications. Maximum-spread interleavers are interleavers whose spread factors achieve the upper bound. An infinite sequence of quadratic permutation polynomials over integer rings that generate maximum-spread interleavers is presented. New properties of permutation polynomial interleavers are investigated from an algebraic-geometric perspective resulting in a new non-linearity metric for interleavers. A new interleaver metric that is a function of both the non-linearity metric and the spread factor is proposed. It is numerically demonstrated that the spread factor has a diminishing importance with the block length. A table of good interleavers for a variety of interleaver lengths according to the new metric is listed. Extensive computer simulation results with impressive frame error rates confirm the efficacy of the new metric. Further, when tail-biting constituent codes are used, the resulting turbo codes are quasi-cyclic.Comment: 25 pages, 11 figures, 6 tables, submitted to IEEE Trans. on Inform. Theor
Recently, the authors introduced an algebraic design framework for space-time coding in flat-fading channels [1]-[4]. In this correspondence, we extend this framework to design algebraic codes for multiple-input multiple-output (MIMO) frequency-selective fading channels. The proposed codes strive to optimally exploit both the spatial and frequency diversity available in the channel. We consider two design approaches: The first uses space-time coding and maximum likelihood decoding to exploit the multi-path nature of the channel at the expense of increased receiver complexity. Within this time domain framework, we also propose a serially concatenated coding construction which is shown to offer a performance gain with a reasonable complexity iterative receiver in some scenarios. The second approach utilizes the orthogonal frequency division multiplexing technique to transform the MIMO multi-path channel into a MIMO flat block fading channel. The algebraic framework [1] is then used to construct space-frequency codes (SFC) that optimally exploit the diversity available in the resulting flat block fading channel. Finally, the two approaches are compared in terms of decoder complexity, maximum achievable diversity advantage, and simulated frame error rate performance in certain representative scenarios. Index Terms-Algebraic space-time codes (STC), diversity, fading channels, multiple transmit and receive antennas, stacking construction. I. INTRODUCTION The first prior work on space-time coding for frequency-selective channels appeared in [5], where it was argued that space-time codes (STCs) designed to achieve a certain diversity advantage in flat-fading channels, will achieve at least the same diversity advantage in frequency-selective fading channels. Similar arguments also appear in [6]. Recent independent work in this area include a space-time block code in [7] and some orthogonal frequency-division multiplexing (OFDM)based designs in [8], [9]. In this correspondence, we design STCs that fully exploit the spatial and frequency diversity available in the channel. The uniqueness of our correspondence is that we attempt to realize this goal by developing an algebraic framework for STC design in such channels. This framework benefits from our earlier work [1], [4] for nonlayered STC design in multiple-input multiple-output (MIMO) flat-fading channels. Codes designed using this framework can achieve guaranteed level of diversity, which includes both space and frequency diversity. The focus on Manuscript
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