Optimizing short stabilizer codes for asymmetric channels

Noise is a major obstacle in the development of practical schemes for quantum computation and communication. Similar to the case of classical communication, this noise can be protected against by employing a code, which provides a means for encoding quantum states prior to transmission and allows for errors to be inferred, and hopefully corrected, by a decoder at the receiver. Unfortunately, designing good codes and decoders is typically a difficult problem. This thesis focuses on developing low-complexity heuristic approaches to three such problems: the design of modified belief propagation decoders for quantum low-density parity-check codes, the design of stabilizer codes for asymmetric channels, and the design of codeword stabilized codes. Quantum low-density parity-check codes are stabilizer codes with low-weight generators. Such codes permit low-complexity decoding via the use of belief propagation, which is an iterative message passing algorithm that takes place on a factor graph defined by the code. However, the performance of such a decoder is limited both by code structure and the degenerate nature of quantum errors. To overcome these limitations, at least in part, a number of modifications to belief propagation are developed. Central among these is the augmented decoder, which in the case of a decoding error, iteratively reattempts decoding using modified factor graph. This heuristic modification simply involves the duplication of a randomly selected subset of the graph’s check nodes, which are in one-to-one correspondence with the code’s stabilizer generators. Across a range of codes, it is shown that the decoders developed perform as well as or better than other modified decoders presented in literature. For a number of channels of physical interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these so-called asymmetric channels, the decoding error rate can be minimized by tailoring the code used to the channel. However, assessing the performance of codes on a given channel is made difficult by the #P-completeness of optimal decoding. To address this complexity, it is shown that the decoding error rate can be accurately approximated using only a small fraction of the possible errors caused by the channel. This approximation is then used to identify a number of cyclic stabilizer codes that perform well on two different asymmetric channels. To further build on this, a heuristic is demonstrated for assessing code performance based on the decoding error rate of an associated classical code. The complexity of calculating this classical error rate is relatively low, and it is shown that it can be used as the basis for a hill-climbing search algorithm. Such searches have yielded a large number of highly performant codes satisfying various structure constraints. The family of codeword stabilized codes encompasses both the stabilizer codes as well as many of the best known nonadditive codes. Constructing a standard form codeword stabilized code is a matter of selecting a simple undirected graph and a binary classical code. This makes designing optimal codes difficult as the number of possible graphs grows exponentially with code length, and the clique search required to construct the classical code is NP-hard. To address the exponential growth of the search space, a heuristic is developed for assessing graphs. This heuristic is then employed by a genetic algorithm that also makes use of a novel crossover operation based on spectral bisection, which is show to be superior to more standard crossover operations. With a graph selected, it is demonstrated that the complexity of the clique search required to construct the associated classical code can be mitigated through the use of a heuristic clique finding algorithm. A number of best known codes are presented that have been found using this approach.

Polar coding is the error correction coding proposed by Arikan. In polar coding, the error rate of the decoding result differs for each bit position owing to the channel polarization. In the case of polar coding with an infinite codeword length, the decoding error rate at each position of the decoded bit sequence is polarized to 0.5 or 0. Therefore, when a codeword of a sufficient length can be adopted, information bits will be decoded without errors by placing the frozen bits to bit positions where the decoding error rates are 0.5 at the transmitter. On the other hand, in the case of polar coding with a finite codeword length, the decoding error rate at each bit position takes a value between 0.5 and 0. Therefore, the decoding errors also occur at the bit positions where the information bits are placed. In this paper, we introduce half-frozen bits into polar coding. Half-frozen bits are the information bits which play roles like frozen bits. Half-frozen bits are placed at positions where errors hardly occur within the frozen bit positions as well as where errors prone to occur within the information bit positions. As half-frozen bits, a part of information bits is used after being applied another error correction coding. This improves the decoding error rates at information bit positions which result in high decoding error rates. Therefore, half-frozen bits improve the overall bit error rate (BER) performance of polar coding. Through computer simulation, we show that the polar coding with introducing half-frozen bits provides a higher error correction performance than the original polar coding.

The energy efficient design of coded automatic-repeat-request (ARQ) systems is studied in this paper. The optimization aims to minimize the energy required for the successfully delivery of one information bit from a transmitter to a receiver. The design is performed by incorporating a wide range of practical system parameters and metrics, such as hardware power consumption, modulation, channel coding, and frame error rate (FER) in the physical layer, and frame length and protocol overhead in the media access control layer. A new log-domain threshold approximation method is proposed to analytically quantify the impacts of the various system parameters on the FER, and the results are used to facilitate the system design. The optimum transmission energy and frame length that minimize the energy per information bit are identified in closed-form expressions as functions of the various practical system parameters. The analytical and simulation results demonstrate that the total energy consumption in a coded ARQ system can be reduced by increasing the transmission energy during one transmission attempt, and significant energy saving as high as 9.5 dB is achieved with the optimum system.

For the design of the QPSK‐TDMA satellite communication system, analyses and experiments of modem systems have been made. When the burst‐mode QPSK signals with white Gaussian noise are input into the demodulation system, the demodulation system including the carrier recovery circuit, is influenced by noise. In this paper, the symbol error rate in the CPSK transmission, the code error rate in the DCPSK transmission, the unique word error rate (UWER) [for PSK transmission for both C and DC] which is important for the frame timing synchronization signal extraction, and occurrence probability of cycle slip in CR as well as its effect on the UWER, have been analyzed. Furthermore, experiments have been made and their results have been compared with the results of the analyses. Also, it has been shown by examples that the modem system of the QPSK‐TDMA system can easily be designed by using the results of the analyses.

Polar codes are the first provable capacity-achieving channel codes. Despite the splendid performance of long Polar codes, short Polar codes have relatively poor performance compared with other modern channel coding schemes (e.g., Turbo codes and LDPC). In this paper, we explore some practical methods to improve the performance of Polar codes with short to moderate codeword lengths. First, we use Reed Solomon (RS) codes as outer codes. With a specific interleaving strategy, we can concatenate multiple RS codes with one frame of Polar codes. Combining a strategy of allocating unequal RS code rates with the concatenation, different levels of protection are assigned based on the error pattern of successive cancellation list (SCL) decoders. Thus, the finite length performance will certainly be enhanced for this encoding scheme. Meanwhile, the memory size that the original SCL decoding procedure requires is reduced, and the increment of overall decoding complexity is small. Additionally, we propose an intra-frame interleaver to further enhance the performance by dispersing errors. Finally, we designed a list decoding scheme for the proposed multi-RS concatenated Polar codes. Depending on the soft information generated by an SCL decoder, we calculated the reliability of each RS symbol and conducted soft RS decoding. So, the overall performance was enhanced under this joint decoding strategy. Simulation results indicate that the bit error rate (BER) performance of short Polar codes can be well improved.

In this paper, we study the resource allocation problem for downlink multicarrier non-orthogonal multiple access systems with short-packet communications (MC-NOMA-SPC). In contrast to long-packet communications in conventional wireless systems, SPC suffers from a transmission rate degradation and an unavoidable decoding error rate. Thus conventional resource allocation based on the Shannon capacity assuming infinite blocklength is no longer optimal. In this paper, we propose to use the effective-throughput as the performance metric to evaluate the tradeoff between the transmission rate and the decoding error rate. Building on that, we jointly optimize the subcarrier assignment, transmission power allocation, and transmission rate adaptation of each user to maximize the total weighted effective-throughput subject to transmission reliability and various practical constraints. Since the formulated problem belongs to a non-convex mixed integer nonlinear programming (MINLP) problem, we develop an efficient algorithm based on the dynamic programming (DP) recursion framework to obtain its optimal solutions. We further propose a low-complexity algorithm based on the principles of block coordinate descent (BCD) and concave-convex procedure (CCCP). Finally, the simulation results show that the proposed low-complexity algorithm can achieve similar performance as that of the optimal solution and outperform the other baseline schemes significantly.

Distributed arithmetic coding (DAC) is similar to syndrome coding, in the sense that message sequences sharing the same interval can be considered a coset of the space of the source sequences, and the codeword is the index of the coset. In this paper, the minimum Hamming distance of cosets is studied and it is proved that such a distance is as small as one. By only allowing the sequences with a large Hamming distance to overlap in the same interval, an improved DAC scheme is proposed. Simulation results show that, for equiprobable memoryless sources, this approach outperforms DAC in terms of decoding error rate at the same coding cost. In addition, at small sequence length, the decoding error rate of the proposed scheme is lower than that of distributed source coding based on low-density parity-check codes for highly correlated sources.

Orthogonal frequency division multiplexing (OFDM) is a type of digital modulation technique which is used in wireless communication systems. It is an efficient multicarrier modulation technique, in which large bandwidth is divided into parallel narrow bands each of which is modulated by different subcarriers. All the subcarriers are orthogonal to each other and hence it reduces the interference among various subcarriers. But OFDM suffers from high peak to average power ratio(PAPR), high complexity in design. The performance parameter of OFDM can be measured in terms of Bit error rate (BER). Lower the BER value, higher is the efficiency of OFDM system. Various methods are used to enhance the BER performance. Channel coding is the one way of enhancing the BER. This paper discusses the performance of OFDM using linear & cyclic block coding for different digital modulation techniques. Bit error rate (BER) is used to measure the performance of OFDM systems.

In this paper a new decoding algorithm for low density parity check (LDPC) codes based on bit flipping (BF) is proposed. In the proposed algorithm, the BF criterion benefits the data correlation between the decoded and received signal in current and previous stages. The proposed algorithm also degrades the decoding probability of one code word to another one in the same codebook. Simulation results confirm our theoretical formulation and show a good enhancement in code error rate and bit error rate of the proposed decoding algorithm compared to the conventional ones.

Convolutional codes are channel codes, which are extensively used in communication systems like GSM (global system for mobile communications) and Interim standard IS-95.We introduce a strategy to present convolutional codes to students learning wireless communication systems, digital communication or similar courses, without using mathematical structures.In this paper, we will discuss our method to implement convolutional encoding and Viterbi decoding to determine the bit error rate.We have compared its performance under different conditions.To exemplify and illuminate our approach, we have selected a communication channel with rate 1/3 convolutional code.Constrain length is three for both Viterbi hard decision decoding method and Viterbi soft decision decoding method.Results suggest that soft decision decoding has at least 3.24 dB improvements in SNR compared to hard decision decoding for same bit error rate.Also by using soft decision Viterbi decoding on AWGN channel, we have examined various convolutional codes for four different constrain lengths and rates.We find that the first code with rate 1/3 code and constrain length 3 has better bit error rate performance with respect to other coding methods.This means, it gives a lower bit error rate for the same value of signal to noise ratio used in other coding schemes.Thus, to achieve the same bit error rate the first code will require a lower SNR.A lower SNR means a lower transmitter power.However, if we use this code the bandwidth requirement is three times larger compared with an un-coded transmission.For the second coding scheme we have an increase in bandwidth by a factor of 2 and for the third and fourth coding scheme the increase factor is 1.5.We believe this approach and its corresponding results will expose students to conceptual and technical aspects of signal encoding and its analysis.

This work studies the performance of a cooperative network which consists of two channel-coded sources, multiple relays, and one destination. To achieve high spectral efficiency, we assume that a single time slot is dedicated to relaying. Conventional network-coded-based cooperation (NCC) selects the best relay which uses network coding to serve the two sources simultaneously. The bit error rate (BER) performance of NCC with channel coding, however, is still unknown. In this paper, we firstly study the BER of NCC via a closed-form expression and analytically show that NCC only achieves diversity of order two regardless of the number of available relays and the channel code. Secondly, we propose a novel partial relaying-based cooperation (PARC) scheme to improve the system diversity in the finite signal-to-noise ratio (SNR) regime. In particular, closed-form expressions for the system BER and diversity order of PARC are derived as a function of the operating SNR value and the minimum distance of the channel code. We analytically show that the proposed PARC achieves full (instantaneous) diversity order in the finite SNR regime, given that an appropriate channel code is used. Finally, numerical results verify our analysis and demonstrate a large SNR gain of PARC over NCC in the SNR region of interest.

Quantum error correction enables the preservation of logical qubits with a lower logical error rate than the physical error rate, with performance depending on the decoding method. Traditional decoding approaches rely on the binarization (“hardening”) of readout data, thereby ignoring valuable information embedded in the analog (“soft”) readout signal. We present experimental results showcasing the advantages of incorporating soft information into the decoding process of a distance-3 (d=3) bit-flip surface code with flux-tunable transmons. We encode each of the 16 computational states that make up the logical state |0L⟩, and protect them against bit-flip errors by performing repeated Z-basis stabilizer measurements. To infer the logical fidelity for the |0L⟩ state, we average across the 16 computational states and employ two decoding strategies: minimum-weight perfect matching and a recurrent neural network. Our results show a reduction of up to 6.8% in the extracted logical error rate with the use of soft information. Decoding with soft information is widely applicable, independent of the physical qubit platform, and could allow for shorter readout durations, further minimizing logical error rates. Published by the American Physical Society 2024

The invention of Fountain codes is a major advance in the field of error correcting codes. The goal of this work is to study and develop algorithms for source and channel coding using a family of Fountain codes known as Raptor codes. From an asymptotic point of view, the best currently known sum-product decoding algorithm for non binary alphabets has a high complexity that limits its use in practice. For binary channels, sum-product decoding algorithms have been extensively studied and are known to perform well. In the first part of this work, we develop a decoding algorithm for binary codes on non-binary channels based on a combination of sum-product and maximum-likelihood decoding. We apply this algorithm to Raptor codes on both symmetric and non-symmetric channels. Our algorithm shows the best performance in terms of complexity and error rate per symbol for blocks of finite length for symmetric channels. Then, we examine the performance of Raptor codes under sum-product decoding when the transmission is taking place on piecewise stationary memoryless channels and on channels with memory corrupted by noise. We develop algorithms for joint estimation and detection while simultaneously employing expectation maximization to estimate the noise, and sum-product algorithm to correct errors. We also develop a hard decision algorithm for Raptor codes on piecewise stationary memoryless channels. Finally, we generalize our joint LT estimation-decoding algorithms for Markov-modulated channels. In the third part of this work, we develop compression algorithms using Raptor codes. More specifically we introduce a lossless text compression algorithm, obtaining in this way competitive results compared to the existing classical approaches. Moreover, we propose distributed source coding algorithms based on the paradigm proposed by Slepian and Wolf.

With the rapid development of the Internet, the Internet of Things, big data, cloud computing, virtual reality and other applications are emerging one after another. New applications have put forward higher requirements for satellite communications. The use of appropriate channel coding technology can optimize the transmission system. This article aims to discuss the satellite channel coding and modulation technology based on cloud computing, to achieve low error rate and strong anti-noise ability of the coding and modulation system under low signal-to-noise ratio. This paper proposes the application of high-performance LDPC codes to QPSK modulation systems commonly used in satellite communications to form a complete LDPC coded QPSK modulation baseband processing system, and comprehensive simulation and verification of the results show that the coded modulation system is better than uncoded The modulation system has a very high gain, the coded modulation scheme has a gain of 12 dB compared to the uncoded modulation system at a bit error rate of 10−7. The coded modulation system designed and implemented in this paper has strong anti-noise ability, low bit error rate and good stability. It is suitable for low signal-to-noise ratio environment such as satellite communication.

This paper presents a low-power energy-harvesting circuit with error-correction demodulation for radio-frequency identification (RFID) application. This circuit uses the new structure to demodulate ASK signals and includes the power unit with a wake-up circuit and a power-on-reset controller. The energy-harvesting circuit proposed by this paper can substantially decrease the capacitance used in the demodulator circuit and the wake-up circuit, and the capacitance is 14 pF only. Moreover, a detection circuit and a correcting circuit are adopted to enhance bit error rate of decoding. The chip was implemented in CMOS 0.18 μm technology. The sensitivity of this work is −13 dBm. The pulse width error can be reduced to less than 1% to fit the EPC Gen-2 standard by error-correction demodulator.