Decoding algorithms with reliability information are of considerable interest in error control coding. Such information when available and when properly used increases the error correcting capability of the decoder. We find decoding algorithms that use soft decision with optimal error correcting capability for fixed constraints on decoder complexity.>
Decoding algorithms with reliability information are of considerable interest in error control coding. Such information when available and when properly used increases the error correcting capability of the decoder. We find decoding algorithms that use soft decision with optimal error correcting capability for fixed constraints on decoder complexity.>
This paper presents a unified method for evaluating the error-correction radii of many reliability-based soft-decision decoding algorithms for binary linear block codes. Based on this unified method, these decoding algorithms as well as their potential improvements can be compared directly. The error-correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft-decision decoder to the transmitted signal sequence. It is shown that this problem can be formulated as a constrained optimization problem. Although no general closed-form expression is possible, a simple algorithm that always converges to the optimum solution of this optimization problem is presented. Based on these results, the error-correction radii of some well-known reliability-based soft-decision decoding algorithms are revisited and compared.
We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.
In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRR-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among o(K/sup i/) candidates, where K is the dimension of the code, for an order i of reprocessing. At practical bit-error rates, for codes of length N/spl les/128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to t+1 errors with an order of reprocessing of at most 2, where t is the error-correcting capability of the code considered.
The most important thing in any communication system is reliable transmission of data. To transmit the data through noisy channel, a redundancy is added before transmission which is known as channel coding. This transmitted data is decoded at the receiver with the help of soft decision decoding or hard decision decoding. Soft decision decoding can be classified as reliability based and code structure based decoding. Reliability in coding theory implies the probability of decoding the bit correctly. Most reliable bits give correct decoding output while less reliable bits are more prone to create errors. Iterative decoder accepts soft inputs-including a priori values-and delivers soft outputs that can he split into: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. The commonly used iterative decoder is sum product belief propagation algorithm. We are extending this analysis to the min-sum algorithm for polar codes. Using polar codes, the sum-product and min-sum algorithms are compared and simulation results are demonstrated.
Error correctiong codes (ECC) are widly used in applications to correct errors in data transmission over unreliable or noisy communication channels. Recently, two kinds of promising codes attracted lots of research interest because they provide excellent error correction performance. One is non-binary LDPC codes, and the other is polar codes. This dissertation focuses on efficient decoding algorithms and decoder design for thesetwo types of codes.Non-binary low-density parity-check (LDPC) codes have some advantages over their binary counterparts, but unfortunately their decoding complexity is a significant challenge. The iterative hard- and soft-reliability based majority-logic decoding algorithms are attractive for non-binary LDPC codes, since they involve only finite field additions and multiplications as well as integer operations and hence have significantly lower complexity than other algorithms. We propose two improvements to the majority-logic decoding algorithms. Instead of the accumulation of reliability information in the ex-isting majority-logic decoding algorithms, our first improvement is a new reliability information update. The new update not only results in better error performance and fewer iterations on average, but also further reduces computational complexity. Since existing majority-logic decoding algorithms tend to have a high error floor for codes whose parity check matrices have low column weights, our second improvement is a re-selection scheme, which leads to much lower error floors, at the expense of more finite field operations and integer operations, by identifying periodic points, re-selectingintermediate hard decisions, and changing reliability information.Polar codes are of great interests because they provably achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. We propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. Then wepropose to use a recursive channel combination to calculate symbol-wise channel transition probabilities, which lead to symbol decisions. Our proposed recursive channel combination has lower complexity than simply combining bit-wise channel transition probabilities. The similarity between our proposed method and Arıkan's channel transformations also helps to share hardware resources between calculating bit- and symbol-wise channel transition probabilities. To reduce the complexity of the list pruning, atwo-stage list pruning network is proposed to provide a trade-off between the error performance and the complexity of the symbol-decision SCL decoder. Since memory is a significant part of SCL decoders, we also propose a pre-computation memory-saving technique to reduce memory requirement of an SCL decoder.To reduce the complexity of the recursive channel combination further, we propose an approximate ML (AML) decoding unit for SCL decoders. In particular, we investigate the distribution of frozen bits of polar codes designed for both the binary erasure and additive white Gaussian noise channels, and take advantage of the distribution to reduce the complexity of the AML decoding unit, improving the throughput-area efficiency of SCL decoders.Furthermore, to adapt to variable throughput or latency requirements which exist widely in current communication applications, a multi-mode SCL decoder with variable list sizes and parallelism is proposed. If high throughput or small latency is required, the decoder decodes multiple received words in parallel with a small list size. However, if error performance is of higher priority, the multi-mode decoder switches to a serialmode with a bigger list size. Therefore, the multi-mode SCL decoder provides a flexible tradeoff between latency, throughput and error performance at the expense of small overhead.