Showing papers on "Sequential decoding published in 2015"

List Decoding of Polar Codes

Abstract: We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, $L$ decoding paths are considered concurrently at each decoding stage, where $L$ is an integer parameter. At the end of the decoding process, the most likely among the $L$ paths is selected as the single codeword at the decoder output. Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of $L$ . Alternatively, if a genie is allowed to pick the transmitted codeword from the list, the results are comparable with the performance of current state-of-the-art LDPC codes. We show that such a genie can be easily implemented using simple CRC precoding. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit, and then uses a pruning procedure to discard all but the $L$ most likely paths. However, straightforward implementation of this algorithm requires $\Omega (L n^{2})$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. In this paper, we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem: we devise an efficient, numerically stable, implementation of the proposed list decoder that takes only $O(L n \log n)$ time and $O(L n)$ space.

Spatially Coupled LDPC Codes Constructed From Protographs

Abstract: In this paper, we construct protograph-based spatially coupled low-density parity-check (LDPC) codes by coupling together a series of $L$ disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying $L$ , we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary $L$ . We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large $L$ , the BP thresholds on both the binary erasure channel and the binary-input additive white Gaussian noise channel saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output channels with low-complexity BP decoding.

On Computing Nearest Neighbors with Applications to Decoding of Binary Linear Codes

Abstract: We propose a new decoding algorithm for random binary linear codes. The so-called information set decoding algorithm of Prange (1962) achieves worst-case complexity \(2^{0.121n}\). In the late 80s, Stern proposed a sort-and-match version for Prange’s algorithm, on which all variants of the currently best known decoding algorithms are build. The fastest algorithm of Becker, Joux, May and Meurer (2012) achieves running time \(2^{0.102n}\) in the full distance decoding setting and \(2^{0.0494n}\) with half (bounded) distance decoding.

Low-Latency Successive-Cancellation List Decoders for Polar Codes With Multibit Decision

Abstract: Polar codes, as the first provable capacity-achieving error-correcting codes, have received much attention in recent years. However, the decoding performance of polar codes with traditional successive-cancellation (SC) algorithm cannot match that of the low-density parity-check or Turbo codes. Because SC list (SCL) decoding algorithm can significantly improve the error-correcting performance of polar codes, design of SCL decoders is important for polar codes to be deployed in practical applications. However, because the prior latency reduction approaches for SC decoders are not applicable for SCL decoders, these list decoders suffer from the long-latency bottleneck. In this paper, we propose a multibit-decision approach that can significantly reduce latency of SCL decoders. First, we present a reformulated SCL algorithm that can perform intermediate decoding of 2 b together. The proposed approach, referred as 2-bit reformulated SCL ( 2b-rSCL ) algorithm , can reduce the latency of SCL decoder from ( $3{n}-2$ ) to ( $2{n}-2$ ) clock cycles without any performance loss. Then, we extend the idea of 2-b-decision to general case, and propose a general decoding scheme that can perform intermediate decoding of any $2^{K}$ bits simultaneously. This general approach, referred as $\textit {2}^{K}$ -bit reformulated SCL ( ${2}^{K}$ b-rSCL ) algorithm , can reduce the overall decoding latency to as short as ${n}/2^{K-2}-2$ cycles. Furthermore, on the basis of the proposed algorithms, very large-scale integration architectures for 2b-rSCL and 4b-rSCL decoders are synthesized. Compared with a prior SCL decoder, the proposed (1024, 512) 2b-rSCL and 4b-rSCL decoders can achieve 21% and 60% reduction in latency, 1.66 and 2.77 times increase in coded throughput with list size 2, and 2.11 and 3.23 times increase in coded throughput with list size 4, respectively.

Fast List Decoders for Polar Codes

Abstract: Polar codes asymptotically achieve the symmetric capacity of memoryless channels, yet their error-correcting performance under successive-cancellation (SC) decoding for short and moderate length codes is worse than that of other modern codes such as low-density parity-check (LDPC) codes. Of the many methods to improve the error-correction performance of polar codes, list decoding yields the best results, especially when the polar code is concatenated with a cyclic redundancy check (CRC). List decoding involves exploring several decoding paths with SC decoding, and therefore tends to be slower than SC decoding itself, by an order of magnitude in practical implementations. In this paper, we present a new algorithm based on unrolling the decoding tree of the code that improves the speed of list decoding by an order of magnitude when implemented in software. Furthermore, we show that for software-defined radio applications, our proposed algorithm is faster than the fastest software implementations of LDPC decoders in the literature while offering comparable error-correction performance at similar or shorter code lengths.

Layered Exact-Repair Regenerating Codes via Embedded Error Correction and Block Designs

Abstract: A new class of exact-repair regenerating codes is constructed by stitching together shorter erasure correction codes, where the stitching pattern can be viewed as block designs. The proposed codes have the help-by-transfer property where the helper nodes simply transfer part of the stored data directly, without performing any computation. This embedded error correction structure makes the decoding process straightforward, and in some cases the complexity is very low. We show that this construction is able to achieve performance better than space-sharing between the minimum storage regenerating codes and the minimum repair-bandwidth regenerating codes, and it is the first class of codes to achieve this performance. In fact, it is shown that the proposed construction can achieve a nontrivial point on the optimal functional-repair tradeoff, and it is asymptotically optimal at high rate, i.e., it asymptotically approaches the minimum storage and the minimum repair-bandwidth simultaneously.

Block Markov Superposition Transmission: Construction of Big Convolutional Codes From Short Codes

Abstract: A construction of big convolutional codes from short codes called block Markov superposition transmission (BMST) is proposed. The BMST is very similar to superposition block Markov encoding (SBME), which has been widely used to prove multiuser coding theorems. The BMST codes can also be viewed as a class of spatially coupled codes, where the generator matrices of the involved short codes (referred to as basic codes) are coupled. The encoding process of BMST can be as fast as that of the basic code, while the decoding process can be implemented as an iterative sliding-window decoding algorithm with a tunable delay. More importantly, the performance of BMST can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available. Numerical results show that: 1) the lower bounds can be matched with a moderate decoding delay in the low bit-error-rate (BER) region, implying that the iterative sliding-window decoding algorithm is near optimal; 2) BMST with repetition codes and single parity-check codes can approach the Shannon limit within 0.5 dB at the BER of $10^{-5}$ for a wide range of code rates; and 3) BMST can also be applied to nonlinear codes.

Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies

Abstract: The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank–Shor–Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance.

Clusterless decoding of position from multiunit activity using a marked point process filter

Abstract: Point process filters have been applied successfully to decode neural signals and track neural dynamics. Traditionally these methods assume that multiunit spiking activity has already been correctly spike-sorted. As a result, these methods are not appropriate for situations where sorting cannot be performed with high precision, such as real-time decoding for brain-computer interfaces. Because the unsupervised spike-sorting problem remains unsolved, we took an alternative approach that takes advantage of recent insights into clusterless decoding. Here we present a new point process decoding algorithm that does not require multiunit signals to be sorted into individual units. We use the theory of marked point processes to construct a function that characterizes the relationship between a covariate of interest in this case, the location of a rat on a track and features of the spike waveforms. In our example, we use tetrode recordings, and the marks represent a four-dimensional vector of the maximum amplitudes of the spike waveform on each of the four electrodes. In general, the marks may represent any features of the spike waveform. We then use Bayes's rule to estimate spatial location from hippocampal neural activity. We validate our approach with a simulation study and experimental data recorded in the hippocampus of a rat moving through a linear environment. Our decoding algorithm accurately reconstructs the rat's position from unsorted multiunit spiking activity. We then compare the quality of our decoding algorithm to that of a traditional spike-sorting and decoding algorithm. Our analyses show that the proposed decoding algorithm performs equivalent to or better than algorithms based on sorted single-unit activity. These results provide a path toward accurate real-time decoding of spiking patterns that could be used to carry out content-specific manipulations of population activity in hippocampus or elsewhere in the brain.

Hardness of Decoding Quantum Stabilizer Codes

Abstract: In this paper, we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NP-complete, and are appropriate a similar decoding problem for quantum codes is also known to be NP-complete. However, this decoding strategy is not optimal in the quantum setting as it does not consider error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes (previously known to be NP-hard) is in fact computationally much harder than optimal decoding of classical linear codes, it is #P-complete.

Convolutional Codes in Rank Metric With Application to Random Network Coding

Abstract: Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a noncoherent multishot network, where the unknown and time-variant network is used several times. In order to create dependence between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding.

Online Fountain Codes With Low Overhead

Abstract: An online fountain code is defined as a fountain code for which an optimal encoding strategy can be found efficiently given any instantaneous decoding state. This property is important for data distribution in practical networks. In this paper, we formalize the problem of online fountain code construction, and propose new online fountain codes that outperform known ones in having factor 3–5 lower redundancy overhead. The bounding of the code overhead is carried out using the analysis of the dynamics of random-graph processes.

Comparison of Convolutional and Block Codes for Low Structural Delay

Abstract: The performance of short block length low-density parity-check (LDPC) codes (both binary and nonbinary) and convolutional codes is compared under the constraint of tight structural delay constraints. Additionally, we use fundamental bounds on block codes and low rate turbo codes to evaluate our results in a broader context. It turns out that—depending on the code rate and given delay—convolutional codes are able to outperform fundamental lower bounds for block codes, yielding a definite result on the question, which codes are superior in this regime. From a break-even point onward, convolutional codes cannot compete with block codes anymore and nonbinary LDPC codes show the best performance. Turbo codes with a short interleaver length show competitive results.

A Fully-Parallel Turbo Decoding Algorithm

Abstract: This paper proposes a novel alternative to the Logarithmic Bahl-Cocke-Jelinek-Raviv (Log-BCJR) algorithm for turbo decoding, yielding significantly improved processing throughput and latency. While the Log-BCJR processes turbo-encoded bits in a serial forwards-backwards manner, the proposed algorithm operates in a fully-parallel manner, processing all bits in both components of the turbo code at the same time. The proposed algorithm is compatible with all turbo codes, including those of the LTE and WiMAX standards. These standardized codes employ odd-even interleavers, facilitating a novel technique for reducing the complexity of the proposed algorithm by 50%. More specifically, odd-even interleavers allow the proposed algorithm to alternate between processing the odd-indexed bits of the first component code at the same time as the even-indexed bits of the second component, and vice-versa. Furthermore, the proposed fully-parallel algorithm is shown to converge to the same error correction performance as the state-of-the-art turbo decoding algorithm. Owing to its significantly increased parallelism, the proposed algorithm facilitates throughputs and latencies that are up to 6.86 times superior to those of the state-of-the art algorithm, when employed for the LTE and WiMAX turbo codes. However, this is achieved at the cost of a moderately increased computational complexity and resource requirement.

Reduced-Complexity Linear Programming Decoding Based on ADMM for LDPC Codes

Abstract: The Euclidean projection onto check polytopes is the most time-consuming operation in the linear programming (LP) decoding based on alternating direction method of multipliers (ADMM) for low-density parity-check (LDPC) codes. In this letter, instead of reducing the complexity of Euclidean projection itself, we propose a new method to reduce the decoding complexity of ADMM-based LP decoder by decreasing the number of Euclidean projections. In particular, if all absolute values of the element-wise differences between the input vector of Euclidean projection in the current iteration and that in the previous iteration are less than a predefined value, then the Euclidean projection at the current iteration will be no longer performed. Simulation results show that the proposed decoder can still save roughly 20% decoding time even if both the over-relaxation and early termination techniques are used.

Asynchronous Convolutional-Coded Physical-Layer Network Coding

Abstract: This paper investigates the decoding process of asynchronous convolutional-coded physical-layer network coding (PNC) systems. Specifically, we put forth a layered decoding framework for convolutional-coded PNC consisting of three layers: symbol realignment layer, codeword realignment layer, and joint channel-decoding network coding (Jt-CNC) decoding layer. Our framework can deal with phase asynchrony (phase offset) and symbol arrival-time asynchrony (symbol misalignment) between the signals simultaneously transmitted by multiple sources. A salient feature of this framework is that it can handle both fractional and integral symbol misalignments. For the decoding layer, instead of Jt-CNC, previously proposed PNC decoding algorithms (e.g., XOR-CD and reduced-state Viterbi algorithms) can also be used with our framework to deal with general symbol misalignments. Our Jt-CNC algorithm, based on belief propagation, is BER-optimal for synchronous PNC and near optimal for asynchronous PNC. Extending beyond convolutional codes, we further generalize the Jt-CNC decoding algorithm for all cyclic codes. Our simulation shows that Jt-CNC outperforms the previously proposed XOR-CD algorithm and reduced-state Viterbi algorithm by 2 dB for synchronous PNC. For both phase-asynchronous and symbol-asynchronous PNC, Jt-CNC performs better than the other two algorithms. Importantly, for real wireless network experimentation, we implemented our decoding algorithm in a PNC prototype built on the USRP software radio platform. Our experiment shows that the proposed Jt-CNC decoder works well in practice.

Scaling Exponent of List Decoders With Applications to Polar Codes

Abstract: Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their scaling exponent is studied as a function of the list size. In particular, the error probability is fixed, and the tradeoff between the block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under $\rm MAP$ decoding with list size $L$ for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under $\rm MAP$ decoding, although the introduction of a list can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the scaling exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the scaling exponent of the latter remains open.

Performance Comparison of LDPC Block and Spatially Coupled Codes over GF(q)

Abstract: In this paper, we compare the finite-length performance of protograph-based spatially coupled low-density paritycheck (SC-LDPC) codes and LDPC block codes (LDPC-BCs) over GF(q). To reduce computational complexity and latency, a sliding window decoder with a stopping rule based on a soft belief propagation (BP) estimate is used for the q-ary SC-LDPC codes. Two regimes are considered: one when the constraint length of q-ary SC-LDPC codes is equal to the block length of q-ary LDPC-BCs and the other when the two decoding latencies are equal. Simulation results confirm that, in both regimes, (3,6)-, (3,9)-, and (3,12)-regular non-binary SC-LDPC codes can significantly outperform both binary and non-binary LDPC-BCs and binary SC-LDPC codes. Finally, we present a computational complexity comparison of q-ary SC-LDPC codes and q-ary LDPC-BCs under equal decoding latency and equal decoding performance assumptions.

Variable-Length Convolutional Coding for Short Blocklengths With Decision Feedback

Abstract: This paper presents a variable-length decision-feedback coding scheme that achieves high rates at short blocklengths. This scheme uses the reliability-output Viterbi algorithm (ROVA) to determine when the receiver's decoding estimate satisfies a given error constraint. We evaluate the performance of both terminated and tail-biting convolutional codes at average blocklengths less than 300 symbols, using the ROVA and the tail-biting ROVA, respectively. Comparing with recent results from finite-blocklength information theory, simulations for both the BSC and the AWGN channel show that the reliability-based decision-feedback scheme can surpass the random-coding lower bound on throughput for feedback codes at some blocklengths less than 100 symbols. This is true both when decoding after every symbol is permitted and when decoding is limited to a small number of increments. Finally, the performance of the reliability-based stopping rule with the ROVA is compared with retransmission decisions based on CRCs. For short blocklengths where the latency overhead of the CRC bits is severe, the ROVA-based approach delivers superior rates.

Variable-Node-Based Dynamic Scheduling Strategy for Belief-Propagation Decoding of LDPC Codes

Abstract: Among the belief-propagation (BP) decoding algorithms of low-density parity-check (LDPC) codes, the algorithms based on dynamic scheduling strategy show excellent performance. In this letter, we propose a variable-node-based dynamic scheduling decoding algorithm. For the proposed algorithm, the reliability of variable nodes is evaluated based on the log-likelihood ratio (LLR) values and the parity-check equations; then, a more accurate dynamic selection strategy is presented. Simultaneously, the oscillating variable nodes are processed so that the influence of the spread of error messages caused by oscillation are suppressed. In addition, the proposed algorithm updates the same number of messages in one iteration as the original BP decoding algorithm does, which is different from some other dynamic decoding algorithms. Simulation results demonstrate that the proposed algorithm outperforms other algorithms.

Low-latency list decoding of polar codes with double thresholding

Abstract: For polar codes with short-to-medium code length, list successive cancellation decoding is used to achieve a good error-correcting performance. However, list pruning in the current list decoding is based on the sorting strategy and its timing complexity is high. This results in a long decoding latency for large list size. In this work, aiming at a low-latency list decoding implementation, a double thresholding algorithm is proposed for a fast list pruning. As a result, with a negligible performance degradation, the list pruning delay is greatly reduced. Based on the double thresholding, a low-latency list decoding architecture is proposed and implemented using a UMC 90nm CMOS technology. Synthesis results show that, even for a large list size of 16, the proposed low-latency architecture achieves a decoding throughput of 220 Mbps at a frequency of 641 MHz.

A memory efficient belief propagation decoder for polar codes

Abstract: Polar codes have become increasingly popular recently because of their capacity achieving property. In this paper, a memory efficient stage-combined belief propagation (BP) decoder design for polar codes is presented. Firstly, we briefly reviewed the conventional BP decoding algorithm. Then a stage-combined BP decoding algorithm which combines two adjacent stages into one stage and the corresponding belief message updating rules are introduced. Based on this stage-combined decoding algorithm, a memory-efficient polar BP decoder is designed. The demonstrated decoder design achieves 50% memory and decoding latency reduction in the cost of some combinational logic complexity overhead. The proposed decoder is synthesized under TSMC 45nm Low Power CMOS technology. It achieves 0.96 Gb/s throughput with 14.2mm2 area when code length N=216 which reduces 51.5% decoder area compared with the conventional decoder design.

A General Formula for the Mismatch Capacity

Abstract: The fundamental limits of channels with mismatched decoding are addressed. A general formula is established for the mismatch capacity of a general channel, defined as a sequence of conditional distributions with a general decoding metrics sequence. We deduce an identity between the Verdu–Han general channel capacity formula, and the mismatch capacity formula applied to maximum likelihood decoding metric. Furthermore, several upper bounds on the capacity are provided, and a simpler expression for a lower bound is derived for the case of a non-negative decoding metric. The general formula is specialized to the case of finite input and output alphabet channels with a type-dependent metric. The closely related problem of threshold mismatched decoding is also studied, and a general expression for the threshold mismatch capacity is obtained. As an example of threshold mismatch capacity, we state a general expression for the erasures-only capacity of the finite input and output alphabet channel. We observe that for every channel, there exists a (matched) threshold decoder, which is capacity achieving. In addition, necessary and sufficient conditions are stated for a channel to have a strong converse.

A Simple Decoder for Topological Codes

Abstract: Here we study an efficient algorithm for decoding topological codes. It is a simple form of HDRG decoder, which could be straightforwardly generalized to complex decoding problems. Specific results are obtained for the planar code with both i.i.d. and spatially correlated errors. The method is shown to compare well with existing ones, despite its simplicity.

List decoding method for polar code and memory system using the same

Abstract: A list decoding method for a polar code includes generating a tree-type decoding graph for input codeword symbols; the generating a tree-type decoding graph including, generating a decoding path list to which a decoding edge is added based on a reliability of a decoding path, the decoding path list being generated such that, among decoding paths generated based on the decoding edge, decoding paths within a threshold number of critical paths survive within the decoding path list in an order of high likelihood probability, and determining an estimation value, which corresponds to a decoding path having a maximum likelihood probability from among decoding paths of the decoding path list, as an information word.

Binary Systematic Network Coding for Progressive Packet Decoding

Abstract: We consider binary systematic network codes and investigate their capability of decoding a source message either in full or in part. We carry out a probability analysis, derive closed-form expressions for the decoding probability and show that systematic network coding outperforms conventional network coding. We also develop an algorithm based on Gaussian elimination that allows progressive decoding of source packets. Simulation results show that the proposed decoding algorithm can achieve the theoretical optimal performance. Furthermore, we demonstrate that systematic network codes equipped with the proposed algorithm are good candidates for progressive packet recovery owing to their overall decoding delay characteristics.

Successive cancellation decoding of polar codes using stochastic computing

Abstract: Polar codes have emerged as the most favorable channel codes for their unique capacity-achieving property. To date, numerous approaches for efficient decoding of polar codes have been reported. However, these prior efforts focused on design of polar decoders via deterministic computation, while the behavior of stochastic polar decoder, which can have potential advantages such as low complexity and strong error-resilience, has not been studied in existing literatures. This paper, for the first time, investigates polar decoding using stochastic logic. Specifically, the commonly-used successive cancellation (SC) algorithm is reformulated into the stochastic form. Several methods that can potentially improve decoding performance are discussed and analyzed. Simulation results show that a stochastic SC decoder can achieve similar error-correcting performance as its deterministic counterpart. This work can pave the way for future hardware design of stochastic polar codes decoders.

New Scalable Decoder Architectures for Reed–Solomon Codes

Abstract: In this paper, we devise new scalable decoder architectures for Reed–Solomon (RS) codes, comprising three parts: error-only decoding, error-erasure decoding, and their decoding for singly extended RS codes New error-only decoders are devised through algorithmic transformations of the inversionless Berlekamp–Massey algorithm (IBMA) We first generalize the Horiguchi–Koetter formula to evaluate error magnitudes using the error locator polynomial $\Lambda(x)$ and the auxiliary polynomial $B(x) $ produced by IBMA, which effectively eliminates the computation of error evaluator polynomial We next devise an enhanced parallel inversionless Berlekamp–Massey algorithm (ePIBMA) that effectively takes advantage of the generalized Horiguchi–Koetter formula The derivative ePIBMA architecture requires only $2t+1$ ( $t$ denotes the error correction capability) systolic cells, in contrast with $3t$ or more cells of the existing regular architectures based on IBMA or the Euclidean algorithm Moreover, it may literally function as a linear-feedback-shift-register encoder New error-erasure decoders are devised through algorithmic transformations of the inversionless Blahut algorithm (IBA) The proposed split parallel inversionless Blahut algorithm (SPIBA) yields merely $2t+1$ systolic cells, which is the same number as the error-only decoder ePIBMA The task is partitioned into two separate steps, computing the complementary error erasure evaluator polynomial followed by computing error-erasure locator polynomial, both utilizing SPIBA Surprisingly, it has exactly the same number of cells and literally the same complexity and throughput as the proposed error-only decoder architecture ePIBMA; it employs 33% less hardware and at the same time achieves more than twice faster throughput, than the serial architecture IBA we further propose a unified parallel inversionless Blahut algorithm (UPIBA) by incorporating the key virtues of the error-only decoder ePIBMA into SPIBA The complexity and throughput of the rderivative UPIBA architecture are literally the same as ePIBMA and SPIBA, while performing almost equally efficiently as ePIBMA on error-only decoding and as SPIBA on error-erasure decoding UPIBA also inherits the dynamic power saving feature of ePIBMA and SPIBA Indeed, UPIBA renders highly attractive for on-the-fly implementation of error-erasure decoding We finally demonstrate that the proposed decoders, ie, ePIBMA, SPIBA, and UPIBA, can be magically migrated to decode singly extended RS codes, with negligible add-ons, except that an extra multiplexer is added to their critical paths To the author's best knowledge, this is the first time that a high-throughput decoder for singly extended RS codes is explored

Decoding LDPC codes with mutual information-maximizing lookup tables

Abstract: A recent result has shown connections between statistical learning theory and channel quantization In this paper, we present a practical application of this result to the implementation of LDPC decoders In particular, we describe a technique for designing the message-passing decoder mappings (or lookup tables) based on the ideas of channel quantization This technique is not derived from sum-product algorithm or any other LDPC decoding algorithm Instead, the proposed algorithm is based on an optimal quantizer in the sense of maximization of mutual information, which is inserted in the density evolution algorithm to generate the lookup tables This algorithm has low complexity since it only employs 3-bit messages and lookup tables, which can be easily implemented in hardware Two quantized versions of the min-sum decoding algorithm are used for comparison Simulation results for a binary-input AWGN channel show 03 dB and 12 dB gains versus the two quantized min-sum algorithms On the binary symmetric channel also a gain is seen

A Matrix-Theoretic Approach to the Construction of Non-Binary Quasi-Cyclic LDPC Codes

Abstract: This paper presents two simple and very flexible methods for constructing non-binary (NB) quasi-cyclic (QC) LDPC codes. The proposed construction methods have several known ingredients including base array , masking , binary to non-binary replacement , and matrix-dispersion . By proper choice and combination of these ingredients, NB-QC-LDPC codes with excellent performance can be constructed. The constructed codes can be decoded with a reduced-complexity iterative decoding scheme which significantly reduces the hardware implementation complexity.