Finite Codeword Length Research Articles

Secret key rate of continuous-variable quantum key distribution with finite codeword length

Secret key rate of continuous-variable quantum key distribution with finite codeword length

Neural-Network-Based Secure State Estimation Under Energy-Constrained Denial-of-Service Attacks: An Encoding-Decoding Scheme

This paper is concerned with the secure state estimation issue for a class of networked nonlinear systems under energy-constrained denial-of-service (EC-DoS) cyber-attacks and encoding-decoding scheme (EDS). The information transmissions between sensors and the estimator are executed via a bandwidth-limited communication network, on which the EDS is deployed to convert transmitted signals into finite-length codewords for the purpose of improving transmission efficiency. The EC-DoS attacks, whose intention is to jeopardize the network-based signal transmissions by overloading the communication resource, are assumed to occur in an intermittent way with bounded occurrence frequency/durations owing to the inherent energy constraints on the attackers. Considering the worst case of such EC-DoS attacks, a neural-network (NN)-based state estimator is constructed to generate the desired state estimates for the underlying networked nonlinear system. By employing the Lyapunov stability theory, the estimation error dynamics of the system state and the neural-network weight are jointly analyzed within a unified framework. Subsequently, sufficient conditions are obtained for the existence of the desired NN-based state estimator, and then both the desired estimator gain matrix and the NN tuning parameters are characterized. Finally, the validity of our estimation approach is confirmed by an example.

Power Spectra of Constrained Codes With Level-Based Signaling: Overcoming Finite-Length Challenges

In various practical systems, certain data patterns are prone to errors if written or transmitted. In magnetic recording and communication over transmission lines, data patterns causing consecutive transitions that are not sufficiently separated are prone to errors. In Flash memory with two levels per cell, data patterns causing high–low–high charge levels on adjacent cells are prone to errors. Constrained codes are used to eliminate error-prone patterns, and they can also achieve other goals. Recently, we introduced efficient binary symmetric lexicographically-ordered constrained (LOCO) codes and asymmetric LOCO (A-LOCO) codes to increase density in magnetic recording systems and lifetime in Flash systems by eliminating the relevant detrimental patterns. Due to their application, LOCO and A-LOCO codes are associated with level-based signaling. Studying the power spectrum of a random signal with certain properties is principal for any storage or transmission system. It reveals important properties such as the average signal power at DC, the bandwidth of the signal, and whether there are discrete power components at certain frequencies. In this paper, we first modify a framework from the literature in order to introduce a method to derive the power spectrum of a sequence of constrained data associated with level-based signaling. We apply our method to infinitely long sequences satisfying symmetric and asymmetric constraints. Next, we show how to generalize the method such that it works for a stream of finite-length codewords as well, thus demonstrating how to overcome the associated finite-length challenges. We use the generalized method to devise closed forms for the spectra of finite-length LOCO and A-LOCO codes from their transition diagrams. Our LOCO and A-LOCO spectral derivations can be performed for any code length and can be extended to other constrained codes. We plot these power spectra, and discuss various important spectral properties for both LOCO and A-LOCO codes. We also briefly discuss an alternative method for deriving the power spectrum and introduce an idea towards reaching the spectra of self-clocked codes.

Performance Analysis of Practical QC-LDPC Codes: From DVB-S2 to ATSC 3.0

In this paper, a new analysis method is introduced to predict the gap between Shannon limit and successful decoding signal-to-noise ratio threshold (SNR loss) of practical quasi-cyclic low-density parity check (QC-LDPC) codes. The SNR loss of a QC-LDPC code is summed up from three aspects: 1) the base matrix; 2) the number of iterations for decoding; and 3) the finite codeword length. As an example, the SNR loss of DVB-S2 and ATSC 3.0 LDPC codes is analyzed with the proposed method. Results show that a large portion of the SNR loss of DVB-S2 LDPC codes is due to their base matrices, which is significantly improved in latest ATSC 3.0 standard. Furthermore, with the aid of the proposed analysis method, newly designed QC-LDPC codes with raptor-like structure are proposed. Bit-error rate simulations show that the proposed codes are only 0.4 dB away from Shannon limit under binary-input additive white Gaussian noise channel and only 0.65 dB away from Shannon limit under binary-input Rayleigh fading channel, which are comparable with state-of-art design.

Finite Block-Length Analysis of Large-But-Finite MIMO Systems

This letter studies the performance of point-to-point multiple-input-multiple-output systems in the presence of large but finite numbers of transmit and receive antennas. We use some recent results on the achievable rates of finite-length codes to analyze the system performance in the cases with short packets. The results are presented for different fading models. With codewords of finite length, we derive closed-form expressions for the error probability/throughput and determine the minimum number of antennas which are required to satisfy different error probability/throughput requirements. As we show, while the error-constrained required number of antennas is sensitive to the length of short codewords, its effect is negligible for codewords of moderate/long length.

Design of Distributed Protograph LDPC Codes for Multi-Relay Coded-Cooperative Networks

This paper studies protograph low-density parity-check coded cooperation (CC) schemes for two-hop multi-relay systems with $L$ relays over Nakagami- $m$ quasi-static fading (QSF) channels. We propose two CC schemes, namely schemes I and II, with different maximum code rates to satisfy different transmission requirements. We further design a family of distributed rate-compatible root-protograph (RCRP) codes to achieve full diversity in CC-based multi-relay QSF channels. In particular, our RCRP codes with $L+1$ sub-codewords can realize full diversity in scheme I, and our RCRP codes with two sub-codewords can achieve full diversity in scheme II with a maximum-ratio combiner. In addition, we estimate the asymptotic word error rate and bit error rate of our RCRP codes using a generalized protograph extrinsic information transfer algorithm, which is able to characterize the error performance of finite-length codewords accurately. Analysis and simulation show that our RCRP codes can achieve outage-limit-approaching performance in both multi-relay CC architectures. This makes the RCRP coding framework extremely attractive for multi-relay cooperative communication applications with slow-varying fading.

Joint Sum Rate and Error Probability Optimization: Finite Blocklength Analysis

We study the tradeoff between the sum rate and the error probability in downlink of wireless networks. Using the recent results on the achievable rates of finite-length codewords, the problem is cast as a joint optimization of the network sum rate and the per-user error probability. Moreover, we develop an efficient algorithm based on the divide-and-conquer technique to simultaneously maximize the network sum rate and minimize the maximum users' error probability and to evaluate the effect of the codewords length on the system performance. The results show that, in delay-constrained scenarios, optimizing the per-user error probability plays a key role in achieving high throughput.

On the Performance Gap between ML and Iterative Decoding of Finite-Length Turbo-Coded BICM in MIMO Systems

As real-time applications typically require short length code words, this paper analyzes the minimum achievable code word error rate of the given finite-length code words, namely, maximum likelihood (ML) decoding error probability. For coding schemes ML decoding is too complex, the key contribution of this paper is to analytically assess the performance gap between ML decoding and a practical decoding scheme. We analyze the combination of turbo codes and bit-interleaved coded-modulation that is a spectrally efficient coding scheme adopted in 3GPP long term evolution (LTE). In single-input single-output (SISO) systems, it was shown in the literature that turbo decoding delivers near-ML decoding performance. In this paper, we extend the analysis to a multi-input multi-output (MIMO) system. In contrast to the SISO case, the turbo principle-based iterative decoding scheme is subject to appreciable performance loss compared with the ML decoding, even in good MIMO channel conditions. For this reason, we further analyze potential reasons and examine possible improvements. By means of simulation, it is shown that convergence to a non-ML code word rather than non-convergence is the key reason for the observed performance loss in good channel conditions.

Iteration-Aware LDPC Code Design for Low-Power Optical Communications

Recent low-density parity-check (LDPC) codes have shown capacity-approaching performance for various communications systems. However, their promising performance cannot always be obtained due to practical constraints, such as finite codeword length, finite iteration, finite memory, and finite precision. In this paper, we focus on a practical design method of high-performance LDPC codes under a constraint of finite-iteration decoding for low-power optical communications. We introduce an iteration-aware LDPC code design approach, which is based on decoding trajectory in extrinsic information transfer chart analysis. It is demonstrated that an LDPC code designed by the conventional curve-fitting method exhibits nearly 2 dB of penalty, when the maximum number of iterations is limited. The results suggest that the LDPC code should be adaptively changed, e.g., when the number of decoding iterations is decreased to save power consumption. We also extend our design method to a multi-objective optimization concept by taking average degrees into account, so that the threshold and the computational complexity are minimized at the same time. The proposed Pareto-optimal codes can achieve additional 2-dB gain or 50% complexity reduction at maximum, in low-power decoding scenarios.

Finite Block-Length Analysis of Spectrum Sharing Networks Using Rate Adaptation

This paper studies the throughput of spectrum sharing networks utilizing rate adaptation. We use some recent results on the achievable rates of finite block-length codes to analyze the secondary user (SU) throughput with a constraint on the primary user (PU) codeword drop probability. With codewords of finite length, we derive closed-form expressions for the SU activation probability and throughput. The results are obtained in different scenarios with perfect or no channel state information at the SU transmitter. As demonstrated numerically and analytically, using finite-length codewords and rate adaptation, there is considerable potential for the data transmission of unlicensed secondary users under different quality-of-service requirements of the licensed primary users.

Finite Block-Length Analysis of Spectrum Sharing Networks: Interference-Constrained Scenario

This letter studies the interference-constrained performance of spectrum sharing networks using adaptive power allocation. The throughput of the secondary user is derived in different cases with constraints on the primary user average or instantaneous received interference power. The analysis is based on some recent results on the achievable rates of finite-length codes and shows the effect of the codeword length on the performance of spectrum sharing networks. The results indicate that, with codewords of finite length and different primary user's interference constraints, there is potential for spectrum sharing, if the transmission power is properly adapted.

A Generalized Channel Coding Theory for Distributed Communication

This paper presents generalized channel coding theorems for a time-slotted distributed communication system where a transmitter-receiver pair is communicating in parallel with other transmitters. Assume that the channel code of each transmitter is chosen arbitrarily in each time slot. The coding choice of a transmitter is denoted by a code index parameter, which is known neither to other transmitters nor to the receiver. Fundamental performance limitation of the system is characterized using an achievable region defined in the space of the code index vectors. As the codeword length is taken to infinity, for all code index vectors inside the region, the receiver will decode the message reliably, while for all code index vectors outside the region, the receiver will report a collision reliably. A generalized system error performance measure is defined as the weighted sum of probabilities of different types of communication error events. Assume that the receiver chooses an “operation region” and intends to decode the message if the code index vector is inside the operation region. Achievable bounds on the tradeoff between the operation region and the generalized error performance measure are obtained under the assumption of a finite codeword length.

Finite Block-Length Analysis of the Incremental Redundancy HARQ

This letter studies the power-limited throughput of a communication system utilizing incremental redundancy (INR) hybrid automatic repeat request (HARQ). We use some recent results on the achievable rates of finite-length codes to analyze the system performance. With codewords of finite length, we derive closed-form expressions for the outage probabilities of INR HARQ and study the throughput in the cases with variable-length coding. Moreover, we evaluate the effect of feedback delay on the throughput and derive sufficient conditions for the usefulness of the HARQ protocols, in terms of power-limited throughput. The results show that, for a large range of HARQ feedback delays, the throughput is increased by finite-length coding INR HARQ, if the sub-codeword lengths are properly adapted.

Concatenated coding and hybrid automatic repeat request for wiretap channels

In this study, the authors propose an equivocation scheme for wiretap channels, which is composed of bit-extension mapping, coset coding and hybrid automatic repeat request (HARQ). The inner bit-extension code and outer coset code are used for equivocation of a wiretapper channel, whereas the HARQ scheme is to mitigate noisy errors in a main legitimate channel. These concatenated codes and HARQ are effective and practical for various channel conditions. The average equivocation and the probability of causing imperfect secrecy are analysed for finite codeword lengths. As a function of channel conditions, they investigate the block error rate at the legitimate receiver and the information leakage to the wiretapper. From simulation results, they further determine the minimum requirements of code design for some target values of the `residual' block error rate and information leakage at maximum retransmission.

On Achievable Rates for MIMO Systems with Imperfect Channel State Information in the Finite Length Regime

New achievable rates for arbitrary finite codeword length and probability of error are presented and analyzed for MIMO systems under both ergodic and non-ergodic scenarios. For the non-ergodic scenario, achievable rates are found under a user specified outage probability.

On the Diversity-Multiplexing Tradeoff of Amplify-and-Forward Half-Duplex Relaying

In this paper, an amplify-and-forward (AF) two-path half-duplex relaying scheme is considered in which one of the relays additionally performs inter-relay interference cancellation. We first generalize lower bounds for the diversity-multiplexing tradeoff (DMT) of an arbitrary block lower triangular channel matrix. We then characterize the diversity-multiplexing tradeoff for the AF two-path relaying scheme and show that the DMT achieves the multiple-input single-output (MISO) upper bound. The analysis also demonstrates that, with a careful choice of the coding strategy, the DMT of this scheme is achievable for finite codeword lengths. We then propose using an equivalent linear space time code at the source, which does not require any form of channel state information, as a simple and effective coding strategy to achieve the full DMT of the scheme. From the DMT perspective, the proposed AF two-path relaying with the equivalent linear space time coding outperforms existing schemes. Our analysis is then extended to the slotted-amplify-and-forward (SAF) scheme with multiple relays, where we provide a stronger result by deriving the DMT while taking into account inter-relay interference.

Error Performance of Channel Coding in Random-Access Communication

A new channel coding approach was proposed in [1] for random multiple access communication over the discrete-time memoryless channel. The coding approach allows users to choose their communication rates independently without sharing the rate information among each other or with the receiver. The receiver will either decode the message or report a collision depending on whether reliable message recovery is possible. It was shown that, asymptotically as the codeword length goes to infinity, the set of communication rates supporting reliable message recovery can be characterized by an achievable region which equals Shannon's information rate region possibly without a convex hull operation. In this paper, we derive achievable bounds on error probabilities, including the decoding error probability and the collision miss detection probability, of random multiple access systems with a finite codeword length. Achievable error exponents are obtained by taking the codeword length to infinity.

Properties of quantum gain of coding with information criterion by binary linear codes

AbstractIn quantum information theory, superadditivity in the capacity of a quantum channel occurs as a special property. We present a method of calculating the mutual information analytically for binary linear codes by using square‐root measurement as the decoding process. Many examples of codes showing the existence of superadditivity in capacity have been given in the past, but the scope of the method was not clarified. In the present paper, we show that the method can be applied to any binary linear codes. We also show that the quantum channel capacity is almost fully attained in a finite codeword length by using approximation to simplex code. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 163(4): 48–57, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20646

Properties of Quantum Gain of Coding with Information Criterion by Binary Linear Codes

In quantum information theory, there is superadditivity in capacity of a quantum channel as special property. We showed the calculating method of mutual information analytically for binary linear codes by using Square-root measurement as decoding process. Therefore, many examples of codes which shows existence of superadditivity in capacity have been given. But the scope of method was not clarified. In the present paper, we show the method can be applied to any binary linear codes. Moreover, we show quantum channel capacity is almost attained in finite codeword length by using approximation to simplex code.

Capacity-approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes

We design multilevel coding (MLC) and bit-interleaved coded modulation (BICM) schemes based on low-density parity-check (LDPC) codes. The analysis and optimization of the LDPC component codes for the MLC and BICM schemes are complicated because, in general, the equivalent binary-input component channels are not necessarily symmetric. To overcome this obstacle, we deploy two different approaches: one based on independent and identically distributed (i.i.d.) channel adapters and the other based on coset codes. By incorporating i.i.d. channel adapters, we can force the symmetry of each binary-input component channel. By considering coset codes, we extend the concentration theorem based on previous work by Richardson et al. ( see ibid., vol.47, p.599-618, Feb. 2001) and Kavc/spl caron/ic/spl acute/ et al.(see ibid., vol.49, p.1636-52, July 2003) We also discuss the relation between the systems based on the two approaches and show that they indeed have the same expected decoder behavior. Next, we jointly optimize the code rates and degree distribution pairs of the LDPC component codes for the MLC scheme. The optimized irregular LDPC codes at each level of MLC with multistage decoding (MSD) are able to perform well at signal-to-noise ratios (SNR) very close to the capacity of the additive white Gaussian noise (AWGN) channel. We also show that the optimized BICM scheme can approach the parallel independent decoding (PID) capacity as closely as does the MLC/PID scheme. Simulations with very large codeword length verify the accuracy of the analytical results. Finally, we compare the simulated performance of these coded modulation schemes at finite codeword lengths, and consider the results from the perspective of a random coding exponent analysis.