之型码和级联之型码

本文介绍了一种新型的纠错码，称之为“之型码”。之型码可以形成非常简捷的软输入/软输出译码规则，我们在Max-Log-MAP(MLM)译码方法的基础上提出了一种译码规则，这一译码规则的计算复杂度为每次迭代计算单位信息比特大约需要20次加法运算操作。在仿真实验中，我们用最优译码器和更简捷的次最优译码器进行译码时，其性能在误比特率为10^-5处分别距香农理论极限仅0.9dB和1.4dBH。此外，上述码字与2维turbo码相比具有更低的误码基底值（error floor）。     

Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in/soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP (MLA) formulation, which requires a total of only 20 addition-equivalent operations per information bit, per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65 536, yields a bit error rate (BER) of 10^-5 at 0.9 dB and 1.3 dB away from the Shannon limit by optimal (APP) and low-cost suboptimal (MLA) decoders, respectively. A union bound analysis of the bit error probability of the zigzag code is presented. It is shown that the union bounds for these codes can be generated very efficiently. It is also illustrated that, for a fixed interleaver size, the concatenated code has increased code potential as the number of constituent encoders increases. Finally, the analysis shows that zigzag codes with four or more constituent encoders have lower error floors than comparable turbo codes with two constituent encoders     

Quantum codes from concatenated algebraic-geometric codes

We apply Steane's enlargement of the Calderbank-Shor-Steane (CSS) codes and additive codes over F/sub 4/ to concatenated algebraic-geometric codes to construct many good quantum codes with fewer restrictions on the parameters compared to some known quantum codes. Some of the quantum codes we have constructed are either optimal or have parameters as good as the best known codes, while some have parameters better than those obtained from other known constructions.     

On decoding concatenated codes

The correcting properties of concatenated codes with parallel decoding over an additive channel are investigated. The ith inner decoder's output is a codeword if the Euclidean distance between the received vector and some codeword is less than Δ_i and an erasure otherwise. The outer decoders correct errors and erasures. The error-correcting capability, which is taken to be the minimum length of any noise vector that can cause an error, is obtained for a bank of z inner and outer decoders as a function of the thresholds used. The set of thresholds that maximize the error-correcting capability is also found. It is shown that for a small number of branches, the error-correcting capability is nearly as large as any decoder     

Partially concatenated convolutional codes

We present a new concatenated code construction. The resulting codes can be viewed as intermediate between parallel and serially concatenated convolutional codes. Proper partitioning of the outer code sequence provides a new degree of freedom for code design. Various methods are considered to analyze code properties.     

Space-time codes and concatenated channel codes for wirelesscommunications

Following a brief historical perspective on channel coding, an introduction to space-time block codes is given. The various space-time codes considered are then concatenated with a range of channel codecs, such as convolutional and block-based turbo codes as well as conventional and turbo trellis codes. The associated estimated complexity issues and memory requirements are also considered. These discussions are followed by a performance study of various space-time and channel-coded transceivers. Our aim is first to identify a space-time code/channel code combination constituting a good engineering tradeoff in terms of its effective throughput, bit-error-rate performance, and estimated complexity. Specifically, the issue of bit-to-symbol mapping is addressed in the context of convolutional codes (CCs) and convolutional coding as well as Bose-Chaudhuri-Hocquenghem coding-based turbo codes in conjunction with an attractive unity-rate space-time code and multilevel modulation is detailed. It is concluded that over the nondispersive or narrow-band fading channels, the best performance versus complexity tradeoff is constituted by Alamouti's twin-antenna block space-time code concatenated with turbo convolutional codes. Further comparisons with space-time trellis codes result in similar conclusions     

High-rate recursive convolutional codes for concatenated channel codes

This letter presents the results of the search for optimum punctured recursive convolutional codes (RCCs) of rate k/k+1, for k=2,...,8, suitable for concatenated channel codes whose constituent encoders are recursive, systematic convolutional codes. The mother codes that are punctured are rate-1/2 RCCs proposed for use in parallel and/or serial concatenation schemes. Extensive tables of systematic and nonsystematic puncturing patterns, optimized relative to various objective functions suitable for concatenated channel codes, are presented for several mother codes.     

Soft information in concatenated codes

A new soft decision criterion is derived. This is useful in communication systems where the combination of modulation, interleaving, and error correction is used. The new algorithm is attractive to implement since traceback techniques can be used and only a little extra memory is required to obtain the soft information associated with every symbol. Furthermore, the operations involved are not recursive and the execution is time independent of the received signal quality. The performance of the new algorithm is typically within 0.25-0.1 dB (bit error rate (BER)=10^-1) of that of a significantly more complex algorithm reported previously     

Performance of concatenated channel codes and orthogonal space-time block codes

In this paper we analyze the performance of an important class of MIMO systems that of orthogonal space-time block codes concatenated with channel coding. This system configuration has an attractive combination of simplicity and performance. We study this system under spatially independent fading as well as correlated fading that may arise from the proximity of transmit or receive antennas or unfavorable scattering conditions. We consider the effects of time correlation and present a general analysis for the case where both spatial and temporal correlations exist in the system. We present simulation results for a variety of channel codes, including convolutional codes, turbo codes, trellis coded modulation (TCM), and multiple trellis coded modulation (MTCM), under quasi-static and block-fading Rayleigh as well as Rician fading. Simulations verify the validity of our analysis.     

Performance of parallel concatenated space-time codes

We study a channel turbo-coding scheme that consists of parallel concatenated systematic space-time codes and is referred to as turbo space-time coded modulation (turbo-STCM). The scheme features full rate and simulation results shows that it also provides full diversity. Performance with recursive versus nonrecursive space-time constituent codes is investigated. The advantage of recursive component codes is demonstrated by simulations for a four state 4-PSK turbo-STCM scheme operating over a Rayleigh block-fading channel. It is also shown that the turbo-STCM performs better than conventional space-time codes of similar complexity     

Simultaneous zero-tailing of parallel concatenated codes

In a parallel concatenated convolutional code, an information sequence is encoded by a convolutional encoder, and an interleaved version of the information sequence is encoded by another convolutional encoder. We discuss the situation in which we require both convolutional encoders to end in the all-zero state. To do so, we have to split an information word in two parts. One part contains the true information bits, and the second part contains the so-called tail bits, which are special bits with values computed such that both encoders end in the all-zero state. Depending on the interleaver, a different number of tail bits are needed. By using a constructive method, we give a characterization of all interleavers for a prescribed number of tail bits. We explain the method of encoding. In addition, simulations have been carried out to investigate the performance of codes resulting from simultaneous zero-tailing. This shows that simultaneous zero-tailing is similar in performance as compared to previously known zero-tailing methods (but with fewer trellis termination bits) and that it is better than zero-tailing just one of the encoders.     

Design of parallel concatenated convolutional codes

A parallel concatenated convolutional coding scheme consists of two constituent systematic: convolutional encoders linked by an interleaver. The information bits at the input of the first encoder are scrambled by the interleaver before entering the second encoder. The codewords of the parallel concatenated code consist of the information bits followed by the parity check bits of both encoders. Parallel concatenated codes (turbo codes), decoded through an iterative decoding algorithm of relatively low complexity, have been shown to yield remarkable coding gains close to theoretical limits. We characterize the separate contributions that the interleaver length and constituent codes give to the overall performance of the parallel concatenated code, and present some guidelines for the optimal design of the constituent convolutional codes     

New ARQ protocols using concatenated codes

Two automatic-repeat-request (ARQ) protocols using a concatenated coding scheme are described. The structure, introduced in a codeword of a concatenated coding scheme, is used to improve the performance of ARQ protocols, especially for high error rates in the communication channel. The performance of the scheme described herein is derived through theoretical analysis. The results show that the proposed scheme outperform other similar ARQ protocols     

Superimposed concatenated codes (Corresp.)

A family of codes of lengthn=q^{s+l}over GF(q), with2s leq qare presented which are constructed by superimposing concatenated codes on a concatenated code. The raterand the distance ratiodeltaof the new codes satisfy the relationr=1-delta+delta ln (delta)for sufficiently large values ofnandq/s. The new codes are superior to the comparable Bose-Chaudhuri-Hocquenghem (BCH) codes, forsgeq 3, in the sense that they contain more codewords. An asymptotically good code constructed using these new codes has a distance ratio greater than those of other asymptotically good codes known to the authors for rates smaller than 0.007.     

Analysis and design of parallel concatenated Gallager codes

The convergence characteristics of parallel concatenated Gallager codes using the Gaussian approximation of extrinsic information are analysed. A design strategy is introduced to select the component low-density parity-check codes effectively     

New classes of binary codes constructed on the basis of concatenated codes and product codes

We present new classes of binary codes that are constructed on the basis of concatenated codes and product codes. We discuss the random-error-correction capabilities of these codes. Some examples of the codes for the correction of random errors are given which have at least as many codewords as the best codes previously known (to the authors) with the same minimum distance and same number of check symbols. The burst-error-correction capabilities of the codes are also discussed. Several examples of the codes for the correction of both random errors and burst errors are given. A decoding algorithm for the codes is also described.     

Design of punctured serially concatenated convolutional codes

An effective algorithm for the design of punctured serially concatenated convolutional codes (SCCCs) is proposed. The algorithm is based on the density evolution technique, and its main goal is to design a code matching the outer and the inner encoder in order to reduce the bit error rate (BER) in the SNR operating region ranging from the waterfall to the error floor of the designed SCCCs. The concepts are illustrated for some specific SCCC schemes. Finally, simulation results and comparisons with other approaches proposed in the literature confirm the effectiveness of the proposed algorithm.     

Multiple serial and parallel concatenated single parity-check codes

Single parity-check (SPC) codes are applied in both parallel and serial concatenated structures to produce high-performance coding schemes. The number of concatenations or stages, M, is increased to improve system performance at moderate-to-low bit-error rates without changing the overall code parameters (namely, code rate and code block length). Analytical bounds are presented to estimate the performance at high signal-to-noise ratios. The SPC concatenated codes are considered with binary phase-shift keying and with 16-quadrature amplitude modulation bit-interleaved coded modulation on the additive white Gaussian noise channel and the independent Rayleigh fading channel. Simulations show that the four-stage serial or parallel concatenated SPC codes can, respectively, outperform or perform as well as 16-state turbo codes. Furthermore, decoding complexity is approximately 9-10 times less complex than that of 16-state turbo codes. The convergence behavior of both serial and parallel concatenated SPC codes is also discussed.