Sequential Decoding Algorithms Research Articles

Single-Shot Decoding of Good Quantum LDPC Codes

Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

Relaxed Decoding of Polar Codes With Large Kernels

A simplified decoding method for polar codes with large kernels is presented. The proposed approach consists in decoder-side substitution of some submatrices of the polarizing transformation with matrices, which admit simpler evaluation of log-likelihood ratios. This approach enables complexity reduction for the successive cancellation, list and sequential decoding algorithms.

Efficient Implementation of Density Evolution for Punctured Polar Codes

Polar codes asymptotically achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under low-complexity sequential decoding algorithms such as successive cancellation decoding. However, in their original formulation, the block length of polar codes is limited to integer powers of the dimension of the underlying polarization kernel used, thus imposing strict constraints on possible application scenarios. While leeway in the choice of kernel or concatenation with other codes mitigates this drawback to a certain extent, puncturing presents a promising approach to specify the target length of a polar code with much greater flexibility. In this paper, we present an efficient implementation of the construction of punctured polar codes based on density evolution, a crucial tool in the construction of both regular, i.e., unpunctured, as well as punctured polar codes. Our implementation of density evolution covers the construction of both regular and punctured polar codes and allows for treating the construction of both code classes in a unified framework. Using our implementation, we achieve substantial reductions in the number of density convolutions necessary for the construction of punctured polar codes and obtain tight upper bounds on the block error rates.

Multilevel spatial modulation

In this paper, a new Multilevel Spatial Modulation technique is proposed. It combines computationally efficient multilevel oding and spatial modulation based on trellis codes to increase coding gain, diversity gain, and bandwidth efficiency. The trellis complexity of the single-stage system increases exponentially, whereas in the proposed multilevel system the complexity increases linearly. The proposed system is analyzed with optimal Viterbi and suboptimal sequential decoding algorithms. The results show that sequential decoding saves 75% of the computational power with a loss of 2 dB SNR approximately, when compared with optimal Viterbi decoding, over both fast- and slow-fading channel conditions. Since the antenna index is used as a source of information in spatial modulation, the number of antennae required increases with the throughput and packing a large number of antennas make cross-correlation unavoidable. In this paper, a low complexity modified decoding technique is also proposed for the multilevel spatial modulation system, in which the correlated received signals are equally combined and decoded by the multistage decoder using the Viterbi algorithm. This technique exploits the receiver antenna correlation and makes the decoding complexity independent of number of antennas. The simulation results indicate that the proposed low complexity algorithm gives approximately 8–10 dB gain when compared with optimal Viterbi decoder with equivalent computational complexity when the eight highly correlated signals are equally combined. This may be a suitable solution for mobile handsets where size and computational complexity are the major issues.

A Low Power Parallel Sequential Decoder for Convolutional Codes

A novel decoding algorithm having a simple hardware realization is proposed for convolutional codes. The proposed decoder accepts a simple implementation in hardware in terms of area occupancy and power consumption compared to other decoders for convolutional codes such as those based on the Viterbi algorithm (VA). Furthermore, the processing delays due to looking back and forward in a trellis as in sequential decoding algorithms are avoided, which makes the proposed decoder suitable for fast high data rates wireless communication systems. Simulation results show a comparable bit error rate (BER) performance to optimal decoders with a reduction of power consumption of 60% compared to Viterbi decoders.

A new sequential decoding algorithm based on branch metric

A new sequential decoding algorithm with an adjustable threshold and a new method of moving through the decoding tree is proposed. Instead of the path metric of the conventional sequential decoding algorithms, the proposed algorithm uses a branch metric based on maximum-likelihood criterion. Two new parameters, the jumping-back distance and going-back distance, are also introduced. The performance of the algorithm for long constraint length convolutional codes is compared to those of the other sequential decoding algorithms and the Viterbi algorithm. The results show that the proposed algorithm is a good candidate for decoding of convolutional codes due to its fast decoding capability and good bit error rate (BER) performance.

Fano Decoding with Timeout: Queueing Analysis

In mobile communications, a class of variable-complexity algorithms for convolutional decoding known as sequential decoding algorithms is of interest since they have a computational time that could vary with changing channel conditions. The Fano algorithm is one well-known version of a sequential decoding algorithm. Since the decoding time of a Fano decoder follows the Pareto distribution, which is a heavy-tailed distribution parameterized by the channel signal-to-noise ratio (SNR), buffers are required to absorb the variable decoding delays of Fano decoders. Furthermore, since the decoding time drawn by a certain Pareto distribution can become unbounded, a maximum limit is often employed by a practical decoder to limit the worst-case decoding time. In this paper, we investigate the relations between buffer occupancy, decoding time, and channel conditions in a system where the Fano decoder is not allowed to run with unbounded decoding time. A timeout limit is thus imposed so that the decoding will be terminated if the decoding time reaches the limit. We use discrete-time semi-Markov models to describe such a Fano decoding system with timeout limits. Our queuing analysis provides expressions characterizing the average buffer occupancy as a function of channel conditions and timeout limits. Both numerical and simulation results are provided to validate the analytical results.

Feedback and error protection strategies for wireless progressive video transmission

Simple but meaningful models for a mobile radio channel are introduced and a channel-coding system based on high-memory rate-compatible punctured convolutional codes with an appropriate sequential decoding algorithm, the far-end error decoder (FEED), are presented. In combination with puncturing, we devise a method for unequal error protection (UEP) and error localization within a progressively coded source message without any additional error detection code. The FEED-based channel-coding system does not aim to minimize the bit or word error probability, but to delay the first error within a data frame as far as possible. This channel-coding scheme and the FEED algorithm can be used efficiently with automatic repeat request (ARQ). We present different ARQ strategies. For all forward error-correction (FEC) schemes, bounds are specified which allow the estimation of the performance and appropriate rate allocation. We briefly discuss an efficient fine granular scalable video compression scheme, the progressive texture video codec (PTVC). The proposed scheme generates an embedded bit-stream for each frame and allows reference frames to be adjusted. These source and channel-coding algorithms are used to design several video communication systems based on FEC and ARQ methods. The resulting systems are presented and compared. Performance estimations based on bounding techniques and optimized rate-allocation algorithms are derived and applied. Experimental results show the improvement potential of the proposed systems compared to standard schemes. Video communication over very low bit-rate mobile channels with varying channel conditions is thus made possible.

A maximum-likelihood soft-decision sequential decoding algorithm for binary convolutional codes

We present a trellis-based maximum-likelihood soft-decision sequential decoding algorithm (MLSDA) for binary convolutional codes. Simulation results show that, for (2, 1, 6) and (2, 1, 16) codes antipodally transmitted over the AWGN channel, the average computational effort required by the algorithm is several orders of magnitude less than that of the Viterbi algorithm. Also shown via simulations upon the same system models is that, under moderate SNR, the algorithm is about four times faster than the conventional sequential decoding algorithm (i.e., stack algorithm with Fano metric) having comparable bit-error probability.

A sequential soft-decision decoder for Reed-Solomon codes applied to encoded PSK in Rayleigh-fading channels

A soft-decision stack algorithm with a variable-bias-term branch metric and accurate channel state information estimate is applied to a Reed-Solomon-encoded phase-shift keying (PSK) system in the presence of memoryless Rayleigh fading. To compensate for the variable decoding delay inherent in sequential decoding algorithms, a time-out mechanism is used by the inner decoder: if a time-out occurs before complete decoding of a given block, the decoder declares an erasure. An erasures-and-errors correction decoding algorithm is implemented at the outer decoder to recover any incorrect or incompletely decoded inner code words. Simulation results show that significant improvement over uncoded modulation can be achieved with this approach with moderate cost in decoding complexity.

Expander codes

Using expander graphs, we construct a new family of asymptotically good, linear error-correcting codes. These codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors. We present both randomized and explicit constructions of these codes. Experimental results demonstrate the good performance of the randomly chosen codes.

An inequality on guessing and its application to sequential decoding

Let (X,Y) be a pair of discrete random variables with X taking one of M possible values, Suppose the value of X is to be determined, given the value of Y, by asking questions of the form Is X equal to x? until the answer is Yes. Let G(x|y) denote the number of guesses in any such guessing scheme when X=x, Y=y. We prove that E[G(X|Y)/sup /spl rho//]/spl ges/(1+lnM)/sup -/spl rho///spl Sigma//sub y/[/spl Sigma//sub x/P/sub X,Y/(x,y)/sup 1/1+/spl rho//]/sup 1+/spl rho// for any /spl rho//spl ges/0. This provides an operational characterization of Renyi's entropy. Next we apply this inequality to the estimation of the computational complexity of sequential decoding. For this, we regard X as the input, Y as the output of a communication channel. Given Y, the sequential decoding algorithm works essentially by guessing X, one value at a time, until the guess is correct. Thus the computational complexity of sequential decoding, which is a random variable, is given by a guessing function G(X|Y) that is defined by the order in which nodes in the tree code are hypothesized by the decoder. This observation, combined with the above lower bound on moments of G(X|Y), yields lower bounds on moments of computation in sequential decoding. The present approach enables the determination of the (previously known) cutoff rate of sequential decoding in a simple manner; it also yields the (previously unknown) cutoff rate region of sequential decoding for multiaccess channels. These results hold for memoryless channels with finite input alphabets.

Erasure-free sequential decoding of trellis codes

An erasure-free sequential decoding algorithm for trellis codes, called the buffer looking algorithm (BLA), is introduced. Several versions of the algorithm can be obtained by choosing certain parameters and selecting a resynchronization scheme. These can be categorized as block decoding or continuous decoding, depending on the resynchronization scheme. Block decoding is guaranteed to resynchronize at the beginning of each block, but suffers some rate loss when the block length is relatively short. The performance of a typical block decoding scheme is analyzed, and we show that significant coding gains over Viterbi decoding can be achieved with much less computational effort. A resynchronization scheme is proposed for continuous sequential decoding. It is shown by analysis and simulation that continuous sequential decoding using this scheme has a high probability of resynchronizing successfully. This new resynchronization scheme solves the rate loss problem resulting from block decoding. The channel cutoff rate, demodulator quantization, and the tail's influence on performance are also discussed. Although this paper considers only the decoding of trellis codes, the algorithm can also be applied to the decoding of convolutional codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Performance of sequential decoding of high-rate punctured convolutional codes

The decoding of long memory high-rate punctured convolutional codes by sequential decoding algorithms is investigated. Both the stack and the Fano algorithms have been thoroughly tested through computer simulation with coding rates ranging from R=2/spl sol/3 to R=/spl frac78/. Error and overflow probabilities and variability of decoding effort are similar for both algorithms. With hard quantization, plateaus appear in the cumulatives of decoding effort for both algorithms. Comparing the punctured approach of decoding to the more traditional technique for high-rate codes, it is found that punctured decoders perform a larger number of simpler computations, so that the overall decoding effort is on the average more important for the usual decoder than it is for its punctured counterpart. Finally, computational variability, error and overflow probabilities are no worse for punctured decoders than they are for normal decoders. >

Fast sequential decoding algorithm based on dynamic searching strategy

The authors present a new sequential decoding algorithm based on dynamic searching strategy to improve decoding efficiency. The searching strategy is to exploit both 'sorting' and 'path recording' techniques. By means of sorting, it is possible to identify the correct path in a very fast way and then, by path recording, the bit sequence can be recovered without degrading decoding performance. The authors also develop a conditional resetting scheme to overcome the buffer overflow problem encountered in conventional sequential decoding algorithms. Simulation results show that, for a given code, the decoding efficiency remains the same as that obtained from maximum likelihood function by appropriately selecting sorting length and decoding depth. In addition, this algorithm can easily be mapped onto an area-efficient VLSI architecture to implement long constraint length convolutional decoders for high-speed digital communications.

Sequential decoding of convolutional codes by a compressed multiple queue algorithm

The conventional multiple stack algorithm (MSA) is an-efficient approach for solving erasure problems in sequential decoding. However, the requirements of multiple stacks and large memory make its implementation difficult. Furthermore,the MSA allows only one stack to be in use at a time: the ether stacks will stay idle until the process in that stack is terminated. Thus it seems difficult to implement the MSA with parallel processing technology. A two-stack scheme is proposed to achieve similar effects to the MSA. The scheme greatly reduces the loading for data transfer and I/O complexity required in the MSA, and makes parallel processing possible. An erasure-free sequential decoding algorithm for convolutional codes, the compressed multiple-queue algorithm (CMQA), is introduced, based on systolic priority queue technology, which-can reorder the path metrics in a short and constant time. The decoding speed will therefore be much faster than in traditional sequential decoders using sorting methods. In the CMQA, a systolic priority queue is divided into two queues by adding control signals, thereby simplifying implementation. Computer simulations show that the CMQA outperforms the MSA in bit error rate, with about one-third the memory requirement of the MSA.

Use of interpolating functions in fast state estimation for dynamic systems with missing observations

Stack sequential decoding algorithms and interpolating functions are used for state estimation of nonlinear discrete dynamic systems with missing observations. The proposed scheme is faster and more practical than the estimation scheme using the Viterbi decoding algorithm.

Multidimensional state estimation using stacks for dynamic systems with interference

Stack sequential decoding algorithms of information theory are used to estimate the states of multidimensional dynamic systems with interference. Each component of the observation model is a non-linear function of only one state component, arbitrary random interference, and observation noise. The state model is a non-linear function of states and disturbance noise. States are estimated, component-by-component, in parallel. This results in memory reduction for the implementation of state estimation. The use of a stack sequential decoding algorithm makes state estimation faster, compared to estimation with the Viterbi decoding algorithm. Simulation results have shown that the proposed estimation scheme performs well, whereas the classical estimation schemes cannot, in general, be used to estimate the states of dynamic models with arbitrary random interference.

Fast state estimation scheme for multidimensional dynamic systems

A new state estimation scheme is presented for multidimensional dynamic systems. The state vector is estimated, component-by-component, by the parallel use of the stack sequential decoding algorithms of information theory. A faster state estimation is accomplished with this scheme for longer time intervals with available memory. Simulation results, some of which are presented, have shown that the scheme produces better estimates than the Kalman estimates for some non-linear models with white gaussian observation and disturbance noise, while it produces estimates comparable with the Kalman estimates for linear models with white gaussian noise.

Computationally efficient modified Fano algorithm for sequential decoding

A modification to Fano&apos;s sequential decoding algorithm by way of a new interpretation of the Fano metric and an alteration in the algorithm is suggested. Such a modification is shown to result in a considerable reduction of computational effort of the decoder. With a specific example, the improvement obtained and its implications are discussed.