A linear-time method for contender sifting in breadth-first decoding of error control codes

Convolutional codes are used extensively in numerous applications in order to achieve reliable data transfer. In this paper, the parameters of Convolutional codes are introduced and used to investigate its applications, characteristics and performance when employed as inner and outer code in decoding method. To evaluate the parameters, four schemes of Convolutional codes: non-systematic-recursive, systematic-recursive, non-systematic-non-recursive and systematic-non-recursive; are applied. In addition, the implementation of BCJR (Bahl, Cocke, Jelinek and Raviv) algorithm is discussed by applying APP-SISO (a posteriori probabilities — Soft Input Soft Output) decoding for Convolutional codes. The proposed scheme employing inner and outer decoding of Convolutional codes are analysed and compared by using EXIT characteristic of Convolutional decoders. The channel capacity measurements for different rates inner Convolutional codes are described by using BPSK modulation over AWGN channel. BER performances of outer decoding of Convolutional codes are observed for different schemes which are described the process of area beneath in the inverted EXIT function. This is essential to obtain better performance of BER plot indeed. The result shows that inner decoding of Convolutional codes is the best performance for the EXIT function, particularly for non-systematic-recursive. However, outer decoding of this code has not good performance due to having quite larger area beneath than inner code.

An efficient parallel processing mode is proposed for the implementation of fast forward-only iterative belief propagation decoding of orthogonal convolutional codes based on one-step decoding. In the proposed parallel processing mode, the iterative decoding of orthogonal convolutional codes makes use of the complete set of orthogonal parity-check equations on each information symbol, thus avoiding the error performance loss. The proposed parallel processing allows the speed of iterative decoding to approach that of one-step decoding. Therefore, an advantageous tradeoff can be available between data rate and latency.

A reduced complexity algebraic type algorithm is described for decoding of convolutional codes over GF(q), q>2. It is founded on the same principles as algebraic-sequentia1 decoding. It is proved that for large q, the algorithm has better complexity-reliability tradeoff than the conventional Viterbi algorithm.

Maximum likelihood decoding of convolutional codes by Viterbi decoding method is an effective error correcting technique. However, little study on Viterbi decoding of high rate codes has been done since the number of paths entering each trellis state increases as the code rate increases. This paper presents a new maximum likelihood decoding of convolutional codes of rate (n ‐ 1)/n using a trellis. the trellis is drawn according to the state transitions of the syndrome former of the code, and has only one or two paths entering each state rather than 2n‐1 paths as in the case of conventional trellises. Thus, we can replace the 2n‐1‐ary comparisons at each state needed in the conventional Viterbi decoding for the code of rate (n ‐ 1)/n by binary comparisons. the implementation of maximum likelihood decoders for high rate convolutional codes is simplified greatly by this approach. Moreover, this decoding method shows the same performance as that of conventional Viterbi decoding method from the viewpoint of selecting the maximum likelihood code sequence.

An optimal and a sub-optimal MAP decoder for a tail-biting convolutional code are presented. Necessary conditions for the existence of a tail-biting circle for recursive systematic convolutional (RSC) codes are stated. The performance of a parallel-concatenated convolutional code (PCCC) with tail-biting RSC component codes and trellis terminated RSC component codes are compared.

Decoding using the dual trellis is considered as a potential technique to increase the throughput of soft-input soft-output decoders for high coding rate convolutional codes. However, the dual Log-MAP algorithm suffers from a high decoding complexity. More specifically, the source of complexity comes from the soft-output unit, which has to handle a high number of extrinsic values in parallel. In this paper, we present a new low-complexity sub-optimal decoding algorithm using the dual trellis, namely the dual Max-Log-MAP algorithm, suited for high coding rate convolutional codes. A complexity analysis and simulation results are provided to compare the dual Max-Log-MAP and the dual Log-MAP algorithms. Despite a minor loss of about 0.2 dB in performance, the dual Max-Log-MAP algorithm significantly reduces the decoder complexity and makes it a first-choice algorithm for high-throughput high-rate decoding of convolutional and turbo codes.

This work focuses on convolutional codes defined over Z/sub 2m/ whose underlying generators are binary-based convolutional codes of known performance levels. The algebraic structures describing such codes are finitely generated free modules, and it is shown that the distance structure of these codes are determined by the generating binary codes, in the spirit of previous work on cyclic block codes.

Trellis diagrams have been traditionally exploited for decoding of convolutional codes [197]. Trellis design procedures for linear block codes have been under investigation since 1974 when Bahl, Cocke, Jelinek and Raviv introduced a method of representing the words in an arbitrary linear block code by the path labels in the trellis [5]. It has been found by Wolf J. [204] that an arbitrary linear (n, k,d min ) block code over GF(q), q ≥ 2, can be represented by its trellis diagram which contains N c = n + 1 columns (or depths) and N s ≤ q min {k,n-k} states. He found that such a trellis, also known as a syndrome or a Wolf’s trellis, could be constructed for the soft maximum likelihood trellis decoding (SMLTD) of a code. However, syndrome trellises have a non-regular structure (except for some particular codes, such as cyclic codes) that complicates their practical implementation.

Maximum likelihood soft-decision decoding of linear block codes is addressed in this correspondence. A novel algorithm based on Chase-2 algorithm for the decoding of systematic binary block codes is detailed. A double re-encoding technique in place of the classical algebraic decoding for the computation of the candidate codeword list is the major innovation of the proposed algorithm. This approach has been successfully applied to systematic block codes that have a code rate equal to 1/2 and a parity check matrix composed of an invertible submatrix for the redundancy part. Simulation results show performance close to the optimum maximum likelihood decoding for an excellent tradeoff between BER performance and computational complexity. Then, the challenging issue of designing a decoder for a specific family of short binary block codes, called Cortex codes is also described. Three soft decoders for Cortex codes with lengths equal to 32, 64, and 128 and a code rate equal to 1/2 have been designed. Then, all the decoders were successively implemented onto an field-programmable gate array (FPGA) device. To our knowledge, they are the first efficient digital implementations of Cortex codes.

Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.

In this paper, we study sliding window decoding of braided convolutional codes (BCCs) in the context of a streaming application, where decoder error propagation can be a serious problem. A window extension algorithm and a resynchronization mechanism are introduced to mitigate the effect of error propagation. In addition, we introduce a soft bit-error-rate stopping rule to reduce computational complexity, and the tradeoff between performance and complexity is examined. Simulation results show that, using the proposed window extension algorithm and resynchronization mechanism, the error performance of BCCs can be improved by up to three orders of magnitude with reduced computational complexity.

The research on quantum convolutional codes is aimed at protecting a flow of information over long distance communications. The polynomial representation of a quantum state is defined. Based on the Calderbank-Shor-Steane(CSS)-type construction of quantum codes, a new method for encoding and decoding of CSS-type quantum convolutional codes is presented and corresponding networks are described. The basis state of the code is transformed into the product of an information polynomial by the generator polynomial. Then networks can be realized by operations of polynomial multiplication. Finally, inspired by classical convolutional decoding idea, a quantum Viterbi algorithm with linear complexity is put forward.

We consider windowed decoding of LDPC Convolutional Codes on the Binary Erasure Channel (BEC) to study the trade-off between the decoding latency and the code performance. We make some key observations about regular LDPC Convolutional code ensembles under windowed decoding and give modified constructions of these codes that allow us to efficiently trade-off performance for gains in latency.

Suboptimal decoding of convolutional codes, motivated by the need to deal with large constraint length codes, has in the past been achieved by stack algorithms or sequential decoding, which typically do not produce soft outputs, which may be desirable in some modern iterative decoding frame-works. Motivated by an approximate posterior equalizer, we present a suboptimal decoder which employs a similar decomposition for binary convolutional codes observed in additive white Gaussian noise. This results in mixed-field arithmetic (GF(2) and R). The GF(2) (convolutional code) arithmetic means that the central limit theorem does not apply. Instead, we invoke Gallager's lemma to compute the distribution of the interfering terms. Under various assumptions of independence (resulting in different complexities) the channel posterior probability can be approximated, resulting in a soft-output decoder. Initial results indicate effective performance when dependencies among rows are incorporated. The method may also extend to arbitrary binary linear codes.

A connectivity based, table driven decoder for convolutional codes that is both fast and simple to implement has been devised. The decoding performance can be made to approach that of an optimal decoder. >