Shorter Codeword Research Articles

Encoding Arguments

Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is effective, but the underlying probabilistic machinery can be daunting. “Encoding arguments” provide an alternative presentation in which probabilistic reasoning is encapsulated in a “uniform encoding lemma.” This lemma provides an upper bound on the probability of an event using the fact that a uniformly random choice from a set of size n cannot be encoded with fewer than log 2 n bits on average. With the lemma, the argument reduces to devising an encoding where bad objects have short codewords. In this expository article, we describe the basic method and provide a simple tutorial on how to use it. After that, we survey many applications to classic problems from discrete mathematics and computer science. We also give a generalization for the case of non-uniform distributions, as well as a rigorous justification for the use of non-integer codeword lengths in encoding arguments. These latter two results allow encoding arguments to be applied more widely and to produce tighter results.

Enhanced Successive Cancellation List Decoding of Polar Codes

Polar codes can be efficiently decoded by the successive cancellation list (SCL) algorithm in which the sorter plays an important role. The sorter, however, is time-consuming. We significantly improve this decoding scheme by developing a new non-sorting direct selection method for the SCL algorithm. In the non-sorting strategy, we design a new selector for choosing the $L$ best surviving paths, which is achieved by exploiting the bitwise comparison of the candidate path metrics. The analysis and simulation results show that, for $M$ -bit length path metrics, our direct selection method with time complexity $O(ML)$ ) is superior to the latest simplified bubble sorter and the quick sorting strategy whose time complexity is $O(ML^{2})$ and $O(ML \log L)$ , respectively. In the scenario of short codeword length and large list size, our selector can remarkably reduce the latency and the hardware resource consumption while not assuming that the metrics in the decoding are sorted.

Describing polyhedral tilings and higher dimensional polytopes by sequence of their two-dimensional components

Polyhedral tilings are often used to represent structures such as atoms in materials, grains in crystals, foams, galaxies in the universe, etc. In the previous paper, we have developed a theory to convert a way of how polyhedra are arranged to form a polyhedral tiling into a codeword (series of numbers) from which the original structure can be recovered. The previous theory is based on the idea of forming a polyhedral tiling by gluing together polyhedra face to face. In this paper, we show that the codeword contains redundant digits not needed for recovering the original structure, and develop a theory to reduce the redundancy. For this purpose, instead of polyhedra, we regard two-dimensional regions shared by faces of adjacent polyhedra as building blocks of a polyhedral tiling. Using the present method, the same information is represented by a shorter codeword whose length is reduced by up to the half of the original one. Shorter codewords are easier to handle for both humans and computers, and thus more useful to describe polyhedral tilings. By generalizing the idea of assembling two-dimensional components to higher dimensional polytopes, we develop a unified theory to represent polyhedral tilings and polytopes of different dimensions in the same light.

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.

Large Sets of Orthogonal Sequences Suitable for Applications in CDMA Systems

In this paper, we employ the so-called semi-bent functions to achieve significant improvements over currently known methods regarding the number of orthogonal sequences per cell that can be assigned to a regular tessellation of hexagonal cells, typical for certain code-division multiple-access (CDMA) systems. Our initial design method generates a large family of orthogonal sets of sequences derived from vectorial semi-bent functions. A modification of the original approach is proposed to avoid a hard combinatorial problem of allocating several such orthogonal sets to a single cell of a regular hexagonal network, while preserving the orthogonality to adjacent cells. This modification increases the number of users per cell by starting from shorter codewords and then extending the length of these codewords to the desired length. The specification and assignment of these orthogonal sets to a regular tessellation of hexagonal cells have been solved regardless of the parity and size of $m$ (where $2^m$ is the length of the codewords). In particular, when the re-use distance is $D=4$ the number of users per cell is $2^{m-2}$ for almost all $m$, which is twice as many as can be obtained by the best known methods.

A Robust Beamforming Approach to Guarantee Instantaneous Secrecy Rate

We consider a well-known beamforming approach that induces jamming signals to avoid eavesdropping from a transmitter equipped with an antenna array in this paper. With this beamforming, which is called masked beamforming, in order to guarantee a certain instantaneous secrecy rate, we formulate robust beamforming problems with a constraint that deals with the eavesdropper channel's uncertainty and find the solutions. Unlike a guaranteed ergodic secrecy rate, this guaranteed instantaneous secrecy rate allows us to use codes of short codewords for secure communications with masked beamforming over slow-fading channels. In addition, since the optimal solutions are found in closed form, we do not need to resort to any convex optimization solvers.

A cost- and energy-efficient embedded system design based on instruction execution frequencies

In this paper, an embedded system that takes into account the frequency of the executed instructions to reduce memory space and save energy consumption is proposed. The object codes are split into the frequently executed instructions and the infrequently executed instructions (INFIs) by analyzing the trace files of applications. To reduce the use of memory space, the dictionary-based method is used to compress INFIs. To take into account energy consumption, the top-executed instructions are selected to encode as shorter codewords and wrapped into a pseudo instruction. When a pseudo instruction that contains several codewords is fetched, it can be decompressed to several continuous instructions to reduce the number of memory accesses. In addition, to further reduce energy consumption, a multiple reference table design is proposed to make a pseudo instruction contain more encoded codewords by shortening the length of an encoded instruction. From the simulation results, the proposed design that uses one 256-instruction reference table reduces the energy consumption about 50.4% compared to the dictionary-based method. In addition, to show the improvement of energy consumption for the proposed multiple reference table method over that using one reference table, we also show the simulation results of a design with two 256-instruction reference tables which shows less energy use than a design with one 512-instruction reference table by 12.1%.

A Power Efficient Test Data Compression Method on Count Compatible PRL Coding

A realistic test sets compression method is proposed to effectively reduce test data volume and test application time during system-on-chip (SoC) scan testing, count compatible pattern run-length (CCPRL) coding method counts the consecutive number of the equal to or contrary to the retained patterns, it modifies the compatible code of variable-length pattern run-length (VPRL) coding rules and adds a count code block to replace original rules for increasing compression ratio. Next, the decoder architecture and the state diagram of finite state machine (FSM) are designed. In addition, the power model of test vectors is analyzed, and the power consumption of scanned-in vectors is roughly evaluated. The six largest ISCAS'89 benchmark circuits verify the proposed coding method has a shorter codeword. Experiment results shows that all compression ratios have been increased as much as possible, test data decompression is lossless, less test application time is consumed, yet the peak power and average power consumption of scanned-in test vector needs to be further improved for modern circuit scan testing.

A Hybrid Decoding Scheme for Short Non-Binary LDPC Codes

In this paper, an iterative soft-decision hybrid decoding algorithm for non-binary low-density parity-check (LDPC) codes with short codeword lengths is proposed. The rationale of the approach is to combine the classical belief propagation (BP) iterative LDPC decoding algorithm with the most reliable basis (MRB) decoding algorithm. This allows to achieve significant performance improvements, with a complexity that, for medium/low error rates, is only slightly higher than that of the BP algorithm alone. The performance improvement with respect to pure BP decoding is up to 0.7 dB at codeword error rate (CER) ≈ 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> . Notably, for a fixed MRB order, hybrid decoding achieves a gain up to 0.5 dB at CER ≈ 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> with respect to BP decoding and MRB decoding used alone.

A Comprehensive Test Compression Scheme Based on Precomputed Test Sets

To cope with increasingly rigorous challenges that large scale digital integrated circuit testing is confronted with, a comprehensive compression scheme consisting of test-bit rearrangement algorithm, run-length assignment strategy and symmetrical code is proposed. The presented test-bit rearrangement algorithm can fasten dont-care bits, 0s or 1s in every test pattern on one of its end to the greatest extent so as to lengthen end-run blocks and decrease number of short run-lengths. A dynamical dont-care assignment strategy based on run-lengths can be used to specify the remaining dont-care bits after the test-bit rearrangement, which can decrease run-length splitting and maximize length of run-lengths. The symmetrical code benefits from long run-lengths and only uses 2 4-bit short code words to identify end-run blocks almost as long as a test pattern, and hence the utilization ratio of code words can be heightened. The presented experiment results show that the proposed comprehensive scheme can obtain very higher data compression ratios than other compression ones published up to now, especially for large scale digital integrated circuits, and considerably decrease test power dissipations.

Distributed Joint Source-Channel Coding Using Unequal Error Protection LDPC Codes

This paper presents a general approach to designing distributed joint source channel (DJSC) codes with arbitrary rates for communication of a pair of correlated binary sources over noisy channels. In this approach, both distributed compression and channel error correction are simultaneously achieved by transmitting, for each source, a fraction of the information bits together with the parity bits of a systematic channel code. This approach is shown to be asymptotically optimal, i.e., any rate-pair in the achievable rate-region can be approached as the codeword length is increased. The practical realization of such a code requi res the design of a pair of channel codes with unequal error protection (UEP) properties determined by the inter-source correlation and the channel capacity available to each source. Towards this end, a linear programming based procedure for jointly optimizing the degree profiles of a pair of irregular LDPC codes to achieve the required UEP properties is presented. Experimental results obtained with both binary symmetric channels and binary-input Gaussian channels are presented, which demonstrate that the proposed UEP-DJSC codes can significantly outperform separate source-channel codes, as well as previously reported joint source-channel coding schemes, particularly for short codeword lengths.

Prefix Encoding by Means of the $(2,3)$-Representation of Numbers

A new family of universal self-synchronizable variable-length codes is introduced. This family is not a generalization or improvement of the existing prefix codes, but is based on a new method of integer representation in a mixed base using the radix-2 and the auxiliary radix-3. Upper length bounds for such codes are obtained. The asymptotic estimates for the (2,3)-encoding including the pointwise redundancy are also given. In particular, this implies that the average length of a (2,3)-codeword is shorter than that of Fibonacci code. Elias gamma and delta codes are adopted for the (2,3) -variant with asymptotically shorter codewords as against the original case. Improvement of gamma and delta encoding for all numbers is also presented. One of (2,3)-codes with high density is highlighted as a possible candidate for practical use in data compression. The (2,3) -codes are very simple to construct and they have evident features of strong robustness.

Channel Capacity and Soft-Decision Decoding of LDPC Codes for Spin-Torque Transfer Magnetic Random Access Memory (STT-MRAM)

Spin-torque transfer magnetic random access memory (STT-MRAM) has emerged as a promising non-volatile memory (NVM) technology, featuring compelling advantages in scalability, speed, endurance, and power consumption. In this paper, we focus on large-capacity stand-alone STT-MRAM, and investigate the channel capacity and the viability of applying low-density parity-check (LDPC) codes with soft-decision decoding to correct the memory cell errors and improve the storage density of STT-MRAM. We propose to use LDPC codes with short codeword lengths, with the reliability-based min-sum (RB-MS) algorithm for decoding. Furthermore, we propose to use the capacity-maximization criterion to design the quantizer and minimize the number of quantization bits. Simulation results demonstrate the potential of applying short-block-length LDPC codes with soft-decision decoding to improve the yield and push the scaling limitation of STT-MRAM.

Post-manufacturing, 17-times acceptable raw bit error rate enhancement, dynamic codeword transition ECC scheme for highly reliable solid-state drives, SSDs

A dynamic codeword transition ECC scheme is proposed for highly reliable solid-state drives, SSDs. By monitoring the error number or the write/erase cycles, the ECC codeword dynamically increases from 512 Byte (+parity) to 1 KByte, 2 KByte, 4 KByte…32 KByte. The proposed ECC with a larger codeword decreases the failure rate after ECC. As a result, the acceptable raw bit error rate, BER, before ECC is enhanced. Assuming a NAND Flash memory which requires 8-bit correction in 512 Byte codeword ECC, a 17-times higher acceptable raw BER than the conventional fixed 512 Byte codeword ECC is realized for the mobile phone application without an interleaving. For the MP3 player, digital-still camera and high-speed memory card applications with a dual channel interleaving, 15-times higher acceptable raw BER is achieved. Finally, for the SSD application with 8 channel interleaving, 13-times higher acceptable raw BER is realized. Because the ratio of the user data to the parity bits is the same in each ECC codeword, no additional memory area is required. Note that the reliability of SSD is improved after the manufacturing without cost penalty. Compared with the conventional ECC with the fixed large 32 KByte codeword, the proposed scheme achieves a lower power consumption by introducing the “best-effort” type operation. In the proposed scheme, during the most of the lifetime of SSD, a weak ECC with a shorter codeword such as 512 Byte (+parity), 1 KByte and 2 KByte is used and 98% lower power consumption is realized. At the life-end of SSD, a strong ECC with a 32 KByte codeword is used and the highly reliable operation is achieved. The random read performance is also discussed. The random read performance is estimated by the latency. The latency is below 1.5 ms for ECC codeword up to 32 KByte. This latency is below the average latency of 15,000 rpm HDD, 2 ms.

Multiple LDPC-Encoder Layered Space-Time-Frequency Architectures for OFDM MIMO Multiplexing

This paper is focused on the study of layered space-time-frequency (LSTF) architectures with channel coding for orthogonal frequency division multiplexing (OFDM) multiple-input multiple-output (MIMO) multiplexing systems for high speed wireless communications over a frequency-selective fading channel. In order to achieve the available spatial, temporal and frequency diversities, and also make the system implementation feasible for high speed OFDM MIMO multiplexing, a novel LSTF architecture with multiple channel encoders is proposed with each independently coded layer being threaded in the three-dimensional space-time-frequency transmission resource array. Non-iterative receiver is adopted which consists of list sphere detector and irregular low-density parity-check codes as the constituent codes. Simulation results show that the performance of the proposed multiple-encoder LSTF architecture is very close to that of the conventional single-encoder LSTF where coding is applied across the whole information stream. However, due to the use of multiple parallel encoders/decoders with a shorter codeword length, the proposed LSTF architecture has much lower hardware processing speed and complexity than the conventional LSTF.

Density Evolution for Nonbinary LDPC Codes Under Gaussian Approximation

This paper extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF(q) . We first generalize the definition of channel symmetry for nonbinary inputs to include q-ary phase-shift keying (PSK) modulated channels for prime q and binary-modulated channels for q that is a power of 2. For the well-defined q-ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by (q-1) quantities. Assuming uniform edge weights, we further show that the density of messages computed by the check node decoder (CND) is fully defined by a single number. We then present the approximate density evolution for regular and irregular LDPC codes, and show that the (q-1) -dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q=2p. Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF(3) and GF(4). The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF(q) codes for high-order modulation by using the idea of a channel adapter. We find that codes designed in this fashion outperform those optimized specifically for the binary additive white Gaussian noise (AWGN) channel for a short codewords and a spectral efficiency of 2 bits per channel use (b/cu).

Minimum Energy coding in CDMA Wireless Sensor Networks

A theoretical framework is proposed for accurate comparison of minimum energy coding in coded division multiple access (CDMA) wireless sensor networks (WSNs). Energy consumption and reliability are analyzed for two coding schemes: minimum energy coding (ME), and modified minimum energy coding (MME). A detailed model of consumed energy is described as function of the coding, radio transmit power, the characteristics of the transceivers, and the dynamics of the wireless channel. Since CDMA is strongly limited by multi-access interference, the system model includes all the relevant characteristics of wireless propagation. A distributed and asynchronous algorithm, which minimizes the total energy consumption by controlling the radio power, is developed. Numerical results are presented to validate the theoretical analysis and show under which conditions MME outperforms ME with respect to energy consumption and bit error rate. It is concluded that MME is more energy efficient than ME only for short codewords.

Upper bounds for the performance of turbo-like codes and low density parity check codes

Researchers have investigated many upper bound techniques applicable to error probabilities on the maximum likelihood (ML) decoding performance of turbo-like codes and low density parity check (LDPC) codes in recent years for a long codeword block size. This is because it is trivial for a short codeword block size. Previous research efforts, such as the simple bound technique [20] recently proposed, developed upper bounds for LDPC codes and turbo-like codes using ensemble codes or the uniformly interleaved assumption. This assumption bounds the performance averaged over all ensemble codes or all interleavers. Another previous research effort [21] obtained the upper bound of turbo-like code with a particular interleaver using a truncated union bound which requires information of the minimum Hamming distance and the number of codewords with the minimum Hamming distance. However, it gives the reliable bound only in the region of the error floor where the minimum Hamming distance is dominant, i.e., in the region of high signal-to-noise ratios. Therefore, currently an upper bound on ML decoding performance for turbo-like code with a particular interleaver and LDPC code with a particular parity check matrix cannot be calculated because of heavy complexity so that only average bounds for ensemble codes can be obtained using a uniform interleaver assumption. In this paper, we propose a new bound technique on ML decoding performance for turbo-like code with a particular interleaver and LDPC code with a particular parity check matrix using ML estimated weight distributions and we also show that the practical iterative decoding performance is approximately suboptimal in ML sense because the simulation performance of iterative decoding is worse than the proposed upper bound and no wonder, even worse than ML decoding performance. In order to show this point, we compare the simulation results with the proposed upper bound and previous bounds. The proposed bound technique is based on the simple bound with an approximate weight distribution including several exact smallest distance terms, not with the ensemble distribution or the uniform interleaver assumption. This technique also shows a tighter upper bound than any other previous bound techniques for turbo-like code with a particular interleaver and LDPC code with a particular parity check matrix.

Design of New Family of Multi-length Two-dimensional Codes for Optical CDMA System

A new class of multi-length, constant-weight two-dimensional multi-wavelength orthogonal codes (2D MWOOCs) with large capacity and good correlation properties is constructed followed the multi-length one-dimensional (1D) OOCs. The performance of these new MWOOCs in an OCDMA network with double-services is analyzed. The result shows that media with the shorter codewords performs much better than the media with longer codeword, and OCDMA system with this new multi-length MWOOCs perform much better than the multi-length ID OOCs do. These features allow large capacity multi-media transmission in OCDMA system. Finally, an application of the new MWOOCs to integrate two types of multi-media services are briefly discusses.

Iterative Signal Processing for Coded LSTF Architectures

For high speed orthogonal frequency division multiplexing (OFDM) plus multiple-input multiple-output (MIMO) multiplexing, a new coded layered space-time-frequency (LSTF) architecture with iterative signal processing at the receiver is proposed, where each independent codeword is threaded in the three-dimensional (3D) space-time-frequency transmission resource array. The iterative receiver structure is adopted consisting of a joint minimum-mean-square-error soft-interference-cancellation (MMSE-SIC) detector and the maximum a posteriori (MAP) convolutional decoders. Simulation results show that the proposed LSTF architecture can achieve almost the same performance as the LSTF (i.e., LSTF-a) where coding is applied across the whole information stream. However, due to its structure of multiple parallel lower speed encoders/decoders with shorter codeword length, the proposed LSTF architecture can be more easily implemented than the LSTF-a.