Shorter Codeword Research Articles

Adaptive Segmented Aggregation and Rate Assignment Techniques for Flexible-Length Polar Codes.

Polar codes have garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provable capacity achieving capability. They have been standardized to be used for encoding information on the control channels in 5G wireless networks due to their robustness for short codeword lengths. The conventional approach to generate polar codes is to recursively use 2×2 kernels and polarize channel capacities. This approach however, has a limitation of only having the ability to generate codewords of length Norig=2n form. In order to mitigate this limitation, multiple techniques have been developed, e.g., polarization kernels of larger sizes, multi-kernel polar codes, and downsizing techniques like puncturing or shortening. However, the availability of so many design options and parameters, in turn makes the choice of design parameters quite challenging. In this paper, the authors propose a novel polar code construction technique called Adaptive Segmented Aggregation which generates polar codewords of any arbitrary codeword length. This approach involves dividing the entire codeword into smaller segments that can be independently encoded and decoded, thereby aggregated for channel processing. Additionally a rate assignment methodology has been derived for the proposed technique, that is tuned to the design requirement.

Hybrid Multi Kernel Polar Code Construction and Optimized Decoding Algorithm for 5G Mobile Communications

5G mobile communications faces difficulty in channel coding to satisfy user requirements. This paper evaluates flexibility, reliability and complexity for different channel coding schemes(Turbo codes, Low Density Parity check codes, mother polar codes, Multi Kernel-polar codes) on regular channel AWGN with BPSK modulation of code rate R=1/2. The proposed hybrid Multi Kernel-polar codes will have greater flexibility and reliability for short and large code words. The optimized decoding of Multi Kernel polar codes reduces the number of computations by using repeated LLR’s of sub-kernel matrices.

Finite Blocklength Regime Performance of Downlink Large Scale Networks

Some emerging 5G and beyond use-cases impose stringent latency constraints, which necessitates a paradigm shift towards finite blocklength performance analysis. In contrast to Shannon capacity-achieving codes, the codeword length in the finite blocklength regime (FBR) is a critical design parameter that imposes an intricate tradeoff between delay, reliability, and information coding rate. In this context, this paper presents a novel mathematical analysis to characterize the performance of large-scale downlink networks using short codewords. Theoretical achievable rates, outage probability, and reliability expressions are derived using the finite blocklength coding theory in conjunction with stochastic geometry, and compared to the performance in the asymptotic regime (AR). Achievable rates under practical modulation schemes as well as multilevel polar coded modulation (MLPCM) are investigated. Numerical results provide theoretical performance benchmarks, highlight the potential of MLPCM in achieving close to optimal performance with short codewords, and confirm the discrepancy between the performance in the FBR and that predicted by analysis in the AR. Finally, the meta distribution of the coding rate is derived, providing the percentiles of users that achieve a predefined target rate in a network.

Symbol-Level GRAND for High-Order Modulation Over Block Fading Channels

Guessing random additive noise decoding (GRAND) is a noise-centric decoding method, which is suitable for low-latency communications, as it supports error correction codes that generate short codewords. GRAND estimates transmitted codewords by guessing the error patterns that altered them during transmission. The guessing process requires the testing of error patterns that are arranged in increasing order of Hamming weight. This approach is fitting for binary transmission over additive white Gaussian noise channels. This letter considers transmission of coded and modulated data over block fading channels and proposes a more computationally efficient variant of GRAND, which leverages information on the modulation scheme and the fading channel. In the core of the proposed variant, referred to as symbol-level GRAND, is an expression that approximately computes the probability of occurrence of an error pattern and determines the order with which error patterns are tested. Analysis and simulation results demonstrate that symbol-level GRAND produces estimates of the transmitted codewords faster than the original GRAND at the cost of a small increase in memory requirements.

On the Effects of Kernel Configuration in Multi-Kernel Polar Codes

Polar codes are a relatively new family of linear block codes which have garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provably capacity achieving capability. They have been proposed to be used for encoding information on the control channels in 5G wireless networks due to their robustness for short codeword lengths. The basic approach introduced by Arikan can only be used to generate polar codes of length , . To overcome this limitation, polarization kernels of size larger than (like , , and so on), have already been proposed in the literature. Additionally, kernels of different sizes can also be combined together to generate multi-kernel polar codes, further improving the flexibility of codeword lengths. These techniques undoubtedly improve the usability of polar codes for various practical implementations. However, with the availability of so many design options and parameters, designing polar codes that are optimally tuned to specific underlying system requirements becomes extremely challenging, since a variation in system parameters can result in a different choice of polarization kernel. This necessitates a structured design technique for optimal polarization circuits. We developed the DTS-parameter to quantify the best rate-matched polar codes. Thereafter, we developed and formalized a recursive technique to design polarization kernels of higher order from component smaller order. A scaled version of the DTS-parameter, namely SDTS-parameter (denoted by the symbol in this article) was used for the analytical assessment of this construction technique and validated for single-kernel polar codes. In this paper, we aim to extend the analysis of the aforementioned SDTS parameter for multi-kernel polar codes and validate their applicability in this domain as well.

Multidimensional Fibonacci Coding

Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci sequence increases but at the price of the increased suffix length. We propose a circumvention to this problem by introducing higher-dimensional Fibonacci codes for integer vectors. The resulting multidimensional Fibonacci coding is comparable to the classical one in terms of compression; while encoding several numbers all at once for a shared suffix generally results in a shorter codeword, the efficiency takes a backlash when terms from different orders of magnitude are encoded together. In addition, while laying the groundwork for the new encoding scheme, we provide extensive theoretical background and generalize the theorem of Zeckendorf to higher order. As such, our work unifies several variations of Zeckendorf’s theorem while also providing new grounds for its legitimacy.

27 Open Problems in Kolmogorov Complexity

This formula can be informally read as follows: the ith messagemi brings us log(1=pi) "bits of information" (whatever this means), and appears with frequency pi, so H is the expected amount of information provided by one random message (one sample of the random variable). Moreover, we can construct an optimal uniquely decodable code that requires about H (at most H + 1, to be exact) bits per message on average, and it encodes the ith message by approximately log(1=pi) bits, following the natural idea to use short codewords for frequent messages. This fits well the informal reading of the formula given above, and it is tempting to say that the ith message "contains log(1=pi) bits of information." Shannon himself succumbed to this temptation [46, p. 399] when he wrote about entropy estimates and considers Basic English and James Joyces's book "Finnegan's Wake" as two extreme examples of high and low redundancy in English texts. But, strictly speaking, one can speak only of entropies of random variables, not of their individual values, and "Finnegan's Wake" is not a random variable, just a specific string. Can we define the amount of information in individual objects?

Source Symbol Purging-Based Distributed Conditional Arithmetic Coding.

A distributed arithmetic coding algorithm based on source symbol purging and using the context model is proposed to solve the asymmetric Slepian–Wolf problem. The proposed scheme is to make better use of both the correlation between adjacent symbols in the source sequence and the correlation between the corresponding symbols of the source and the side information sequences to improve the coding performance of the source. Since the encoder purges a part of symbols from the source sequence, a shorter codeword length can be obtained. Those purged symbols are still used as the context of the subsequent symbols to be encoded. An improved calculation method for the posterior probability is also proposed based on the purging feature, such that the decoder can utilize the correlation within the source sequence to improve the decoding performance. In addition, this scheme achieves better error performance at the decoder by adding a forbidden symbol in the encoding process. The simulation results show that the encoding complexity and the minimum code rate required for lossless decoding are lower than that of the traditional distributed arithmetic coding. When the internal correlation strength of the source is strong, compared with other DSC schemes, the proposed scheme exhibits a better decoding performance under the same code rate.

Miscorrection Mitigation for Generalized Integrated Interleaved BCH Codes

The generalized integrated interleaved (GII) codes nest BCH sub-codewords to form codewords of more powerful BCH codes. They can achieve hyper-throughput decoding with excellent error-correcting performance and are among the most suitable codes for next-generation memories. However, the new storage class memories require high code rate and short codeword length. In this case, the sub-codewords have small correction capability. The miscorrections of the sub-codewords lead to much more severe degradation on the GII error-correcting performance compared to the miscorrections of classic BCH codes. This letter investigates the miscorrections in GII decoding. Three low-complexity methods are developed to identify and mitigate the miscorrections by utilizing the nested syndromes, adding one single parity to each sub-codeword, and keeping track of the error locator polynomial degree. Besides, formulas for estimating the miscorrection rates are given. Using the proposed mitigation methods, the actual GII decoding performance becomes very close to that of the case without any miscorrections.

Improving Compression Ratios for Code-Based Test Pattern Compressions through Column-Wise Reordering Algorithms

Chip testing is an effective way to reduce the number of defective or faulty chips that reach the market. However, as large-scale test patterns need to be transmitted into a circuit under test during testing, the transmission time dominates the test application time of the chip testing. Therefore, code-based compression methods are widely used in compressing test patterns because of their capability to reduce the transmission time and save storage space significantly. Current code-based compression methods cannot fully apply the inherent characteristics of test patterns yet. To address this problem, this study proposes two-stage test pattern preprocessing algorithms, thereby improving the efficiency of the code-based compression method. First, we propose a column-wise reordering for Hadamard matrix (CRHM) algorithm, which decomposes a test set consisting of test patterns into a primary component set (PCS) and a residual component set (RCS). The PCS inherits some 1s from the original test set (OTS), and other 1s belong to the RCS. As the number of 1s contained in the RCS is less than that in the OTS, the RCS can obtain a higher code-based compression ratio. The PCS can be generated by an on-chip generator, which does not consume transmission time. Second, we propose a novel column-wise reordering for the RCS (CRRCS) algorithm. The CRRCS solves the new location of each column of the RCS one by one in the list to decrease the entropy of the RCS. The entropy denotes the shortest length of the codeword required for the symbol to be encoded. The smaller entropy value refers to a higher compression ratio. For the sorted RCS, more high-frequency symbols can be replaced by shorter codewords. Experimental results based on seven code-based compression methods show that the proposed algorithms can increase the average compression ratio by a total of 19.91%, and the highest average compression ratio reaches 85.04% for ISCAS’89 benchmark circuits.

Deep Learning-Based Decoding of Linear Block Codes for Spin-Torque Transfer Magnetic Random Access Memory

Thanks to its superior features of fast read/write speed and low power consumption, spin-torque transfer magnetic random access memory (STT-MRAM) has become a promising non-volatile memory (NVM) technology that is suitable for many applications. However, the reliability of STT-MRAM is seriously affected by the variation of the memory fabrication process and the working temperature, and the later will lead to an unknown offset of the channel. Hence, there is a pressing need to develop more effective error correction coding techniques to tackle these imperfections and improve the reliability of STT-MRAM. In this work, we propose, for the first time, the application of deep-learning (DL)-based algorithms and techniques to improve the decoding performance of linear block codes with short codeword lengths for STT-MRAM. We formulate the belief propagation (BP) decoding of linear block code as a neural network (NN) and propose a novel neural normalized-offset reliability-based min-sum (RB-MS) (NNORB-MS) decoding algorithm. We successfully apply our proposed decoding algorithm to the STT-MRAM channel through channel symmetrization to overcome the channel asymmetry. We also propose an NN-based soft information generation method (SIGM) to take into account the unknown offset of the channel. Simulation results demonstrate that our proposed NNORB-MS decoding algorithm can achieve significant performance gain over both the hard-decision decoding (HDD) and the regular RB-MS decoding algorithm, for cases without and with the unknown channel offset. Moreover, the decoder structure and time complexity of the NNORB-MS algorithm remain similar to those of the regular RB-MS algorithm.

A novel reliability-based high performance decoding algorithm for short block length turbo codes

Satellite communication applications use turbo codes as the standard error correcting code due to its Shannon's capacity approaching performance. However, short block length turbo codes exhibit significant performance degradation. This is a limiting factor in applications like transmissions over the telecommand links for satellite communications that involve the use of short length turbo codes. A novel reliability-based turbo decoding algorithm that addresses the performance improvement of short block length turbo codes, is being proposed in this paper. Simulation results show a coding gain of 2.45 dB at BER of 10−3 for short length codewords. The proposed decoding algorithm has low computational complexity compared to the conventional iterative decoding algorithm. The relatively lower computational complexity and the conspicuous improvement in BER performance make the method quite attractive.

New Locally Correctable Codes Based on Projective Reed–Muller Codes

Locally decodable codes and locally correctable codes (LCCs) have several important applications, such as private information retrieval, secure multiparty computation, and circuit lower bounds. Three major parameters are considered in LCCs: query complexity, message length, and codeword length. The most familiar LCCs in the regime of low query complexity are the generalized Reed–Muller (GRM) codes. However, it has not previously been determined whether there exist codes that have shorter codeword lengths than GRM codes with the same query complexity and message length. In this paper, we show that the projective Reed–Muller (PRM) codes are such LCCs for some parameters. The GRM code is specified by the alphabet size $q$ , the number of variables $m$ , and the degree $d$ , where $d\le q-2$ . When $d= q-2$ and $q-1$ is a power of a prime, we prove that there exists a PRM code with shorter codeword length than the GRM code with the same query complexity and message length. We also present for these PRM codes a perfectly smooth local decoder to recover a symbol in a codeword by accessing not more than $q$ symbols at the coordinates of the codeword.

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.

Evaluation of a Joint Detector Demodulator Decoder (JDDD) Performance With Modulation Schemes Specified in the DVB-S2 Standard

With the rapid increase in the growth of digital broadcast technologies, there has been a demand for high data rate, power, and bandwidth efficient transmission, which, today, is aided by the high-performing low-density parity check (LDPC) codes. In this paper, we explore a novel alternative scheme to the LDPC decoder that performs detection, demodulation, and decoding jointly, coined the joint detector demodulator decoder (JDDD). We test the JDDD through simulation using the modulation schemes defined in the DVB-S2 satellite broadcasting standard. The JDDD is a more recently developed algorithm to the sum–product algorithm (SPA) that is used in decoding the LDPC codes. The JDDD is optimal over a modulated additive white Gaussian noise/intersymbol interference (ISI) channel when resources are sufficient, with the main constraint limiting the algorithm being the availability of computing resources. In this paper, we compare the performance of the system using the JDDD against that of the LDPC decoder. The main result is that the JDDD is able to outperform the iterative detector decoder (IDD) at shorter codeword lengths, when resource requirement is smaller, while the increase in computational requirements tends to favor the IDD at longer CWLs.

Test Data Compression with Alternating Equal-Run-Length Coding

This paper presents a new X-filling algorithm for test power reduction and a novel encoding technique for test data compression in scan-based VLSI testing. The proposed encoding technique focuses on replacing redundant runs of the equal-run-length vector with a shorter codeword. The effectiveness of this compression method depends on a number of repeated runs occur in the fully specified test set. In order to maximize the repeated runs with equal run length, the unspecified bits in the test cubes are filled with the proposed technique called alternating equal-run-length (AERL) filling. The resultant test data are compressed using the proposed alternating equal-run-length coding to reduce the test data volume. Efficient decompression architecture is also presented to decode the original data with lesser area overhead and power. Experimental results obtained from larger ISCAS'89 benchmark circuits show the efficiency of the proposed work. The AERL achieves up to 82.05 % of compression ratio as well as up to 39.81% and 93.20 % of peak and average-power transitions in scan-in mode during IC testing.

Short Non-Malleable Codes from Related-Key Secure Block Ciphers

A non-malleable code is an unkeyed randomized encoding scheme that offers the strong guarantee that decoding a tampered codeword either results in the original message, or in an unrelated message. We consider the simplest possible construction in the computational split-state model, which simply encodes a message m as k||Ek(m) for a uniformly random key k, where E is a block cipher. This construction is comparable to, but greatly simplifies over, the one of Kiayias et al. (ACM CCS 2016), who eschewed this simple scheme in fear of related-key attacks on E. In this work, we prove this construction to be a strong non-malleable code as long as E is (i) a pseudorandom permutation under leakage and (ii) related-key secure with respect to an arbitrary but fixed key relation. Both properties are believed to hold for “good” block ciphers, such as AES-128, making this non-malleable code very efficient with short codewords of length |m|+2τ (where τ is the security parameter, e.g., 128 bits), without significant security penalty.

Short Non-Malleable Codes from Related-Key Secure Block Ciphers

A non-malleable code is an unkeyed randomized encoding scheme that offers the strong guarantee that decoding a tampered codeword either results in the original message, or in an unrelated message. We consider the simplest possible construction in the computational split-state model, which simply encodes a message m as k||Ek(m) for a uniformly random key k, where E is a block cipher. This construction is comparable to, but greatly simplifies over, the one of Kiayias et al. (ACM CCS 2016), who eschewed this simple scheme in fear of related-key attacks on E. In this work, we prove this construction to be a strong non-malleable code as long as E is (i) a pseudorandom permutation under leakage and (ii) related-key secure with respect to an arbitrary but fixed key relation. Both properties are believed to hold for “good” block ciphers, such as AES-128, making this non-malleable code very efficient with short codewords of length |m|+2τ (where τ is the security parameter, e.g., 128 bits), without significant security penalty.

Low Complexity Parity Check Code for Futuristic Wireless Networks Applications

The Internet of Things refers to the aptitude of remotely connecting and monitoring anything, anytime, and anywhere via futuristic wireless networks. Due to the unreliable wireless links, broadcast nature of wireless transmissions, interference and noisy transmission channels, frequent topology changes, and the various quality of wireless channel, there are challenges in providing high data rate service, high throughput, high packet delivery ratio, low end-to-end delay, and reliable services. In wireless network and real time application systems, low complexity and shorter codeword length in channel coding scheme are preferred. Consequently, in order to address these challenges, we propose a novel error detection and correction codes called the low-complexity parity-check (LCPC) codes with short codeword lengths for futuristic wireless networks applications. The proposed codes have less complexity and lower memory requirement in comparison to turbo and low-density parity-check (LDPC) codes. Simulation results demonstrated that the proposed LCPC codes outperform the Hamming and Reed–Solomon codes, in addition to the renowned LDPC codes. It offers up to 3-dB coding gain.

Efficient approximate message authentication scheme

An approximate message authentication scheme is a primitive that allows a sender Alice to send a source state to a receiver Bob such that the latter is assured of its authenticity, where the source state is considered as authentic if it only undergoes a minor change. Here, the authors propose an efficient scheme for this problem and prove its security under a rigorous model. Our scheme only needs a lightweight computation cost and hence is very efficient. As the authentication message is transmitted over a noisy channel, we also value the channel efficiency (i.e. the coding rate). For a fixed coding method, this is determined by the admissible decoding bit error probability p e . A larger p e admits a shorter codeword length and hence a larger coding rate. It turns out that the p e can be set to be a significantly large constant (determined by the legal distortion level for the source state). Compared with existing schemes, the advantage in p e is evident.