Outer Code Research Articles

Left dihedral codes over Galois rings [formula omitted

Let D2n=〈x,y∣xn=1,y2=1,yxy=x−1〉 be a dihedral group, and R=GR(p2,m) be a Galois ring of characteristic p2 and cardinality p2m where p is a prime. Left ideals of the group ring R[D2n] are called left dihedral codes over R of length 2n, and abbreviated as left D2n-codes over R. Let gcd(n,p)=1 in this paper. Then any left D2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes Ai and outer codes Ci, where Ai is a cyclic code over R of length n and Ci is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R. Specifically, a generator matrix and basic parameters for each outer code Ci are given. A formula to count the number of these codes is obtained and the dual code for each left D2n-code is determined. Moreover, all self-dual left D2n-codes and self-orthogonal left D2n-codes over R are presented.

Techniques for Improving the Finite Length Performance of Sparse Superposition Codes

Sparse superposition codes are a recent class of codes introduced by Barron and Joseph for efficient communication over the AWGN channel. With an appropriate power allocation, these codes have been shown to be asymptotically capacity-achieving with computationally feasible decoding. However, a direct implementation of the capacity-achieving construction does not give good finite length error performance. In this paper, we consider sparse superposition codes with approximate message passing (AMP) decoding, and describe a variety of techniques to improve their finite length performance. These include an iterative algorithm for SPARC power allocation, guidelines for choosing codebook parameters, and estimating a critical decoding parameter online instead of pre-computation. We also show how partial outer codes can be used in conjunction with AMP decoding to obtain a steep waterfall in the error performance curves. We compare the error performance of AMP-decoded sparse superposition codes with coded modulation using LDPC codes from the WiMAX standard.

Development of the method of features learning and training decision rules for the prediction of violation of service level agreement in a cloud-based environment

We developed the algorithm of learning of the multilayer feature extractor based on ideas and methods of neural gas and sparse encoding, for the problem of prediction of violation of agreement conditions on the service level in a cloud-based environment. Effectiveness of the proposed extractor and autoencoder was compared by the results of physical simulation. It is shown that the proposed extractor requires approximately 1.6 times as few learning samples as the autoencoder for construction of error-free decision rules for learning and test samples. This allows us previously put into effect prediction mechanisms of controlling appropriate cloud-based services. To build up decision rules, it is proposed to use transformation of the space of primary features using computationally efficient operations of comparison and excluding OR for construction in the radial basis of the binary space of secondary features of separate class containers. In this case, for binary feature encoding, it is proposed to use modification of the population algorithm of search for maximum value of the Kullback’s information criterion. Modification implies consideration of compactness of images in the space of secondary features, which allows increasing the gap between distributions of classes and decreasing the negative effect of overfitting. The authors explored dependence of decision accuracy for training and test samples of the system of prediction of violation of SLA conditions on parameters of the feature extractor and those of the classifier. The extractor configuration, acceptable in terms of accuracy and complexity, was selected. In this case, two time windows, which intersect in time by 50 % and read through 50 features, were used at the entrance of the extractor. The first layer of extractor coding contains 30 basis vectors, and the second layer – 20. Thus, the intralayer pooling and non-linearity were formed by concatenation of sparse codes of each window and by continuation of the resulting code twice as much in order to separate positive and negative code components and to transform the resulting code into the vector of sign-positive features.

Magic state distillation with low space overhead and optimal asymptotic input count

We present an infinite family of protocols to distill magic states for T-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T-gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log⁡log⁡1/δ). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.

Constrained Interleaved Coded Spatial Modulation (CICSM)

Constrained interleaved coded spatial modulation (CICSM) is introduced to best map coded bits in spatial modulation (SM) systems to achieve the highest possible minimum symbol Hamming distance and the highest possible minimum squared Euclidean distance. CICSM extends the known constrained interleaved coded modulation technique to spatial modulation. Joint coding and independent coding versions of CICSM are developed with normal mapping and higher-dimensional mapping. Powerful CICSM schemes that perform better than known coded SM schemes in the literature are designed with simple outer block codes.

Nonuniform code concatenation for universal fault-tolerant quantum computing

Using transversal gates is a straightforward and efficient technique for fault-tolerant quantum computing. Since transversal gates alone cannot be computationally universal, they must be combined with other approaches such as magic state distillation, code switching or code concatenation in order to achieve universality. In this paper we propose an alternative approach for universal fault-tolerant quantum computing mainly based on the code concatenation approach proposed in [PRL 112, 010505 (2014)] but in a non-uniform fashion. The proposed approach is described based on non-uniform concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code as well as the 5-qubit code with the 15-qubit Reed-Muller code, which lead to two 49-qubit and 47-qubit codes, respectively. These codes can correct any arbitrary single physical error with the ability to perform a universal set of fault-tolerant gates, without using magic state distillation.

BATS Codes: Theory and Practice

This book discusses an efficient random linear network coding scheme, called BATched Sparse code, or BATS code, which is proposed for communication through multi-hop networks with packet loss. Multi-hop wireless networks have applications in the Internet of Things (IoT), space, and under-water network communications, where the packet loss rate per network link is high, and feedbacks have long delays and are unreliable. Traditional schemes like retransmission and fountain codes are not sufficient to resolve the packet loss so that the existing communication solutions for multi-hop wireless networks have either long delay or low throughput when the network length is longer than a few hops. These issues can be resolved by employing network coding in the network, but the high computational and storage costs of such schemes prohibit their implementation in many devices, in particular, IoT devices that typically have low computational power and very limited storage. A BATS code consists of an outer code and an inner code. As a matrix generalization of a fountain code, the outer code generates a potentially unlimited number of batches, each of which consists of a certain number (called the batch size) of coded packets. The inner code comprises (random) linear network coding at the intermediate network nodes, which is applied on packets belonging to the same batch. When the batch size is 1, the outer code reduces to an LT code (or Raptor code if precode is applied), and network coding of the batches reduces to packet forwarding. BATS codes preserve the salient features of fountain codes, in particular, their rateless property and low encoding/decoding complexity. BATS codes also achieve the throughput gain of random linear network coding. This book focuses on the fundamental features and performance analysis of BATS codes, and includes some guidelines and examples on how to design a network protocol using BATS codes.

Soft Convolutional Codes Decoding Algorithm for Concatenated Codes in Visible Light Communication

In this paper, we propose a new decoding algorithm for soft convolutional codes and concatenated codes in visible light communication (VLC) systems. RLL codes are used to control the dimming in VLC since each codeword has same 1s and 0s, hard RLL decoding and soft RLL decoding are available now. Hard convolutional decoding can be easily combined with RLL decoding based on Viterbi algorithm, however, efficient soft output convolutional decoding combined with RLL codes is worth considering. In our algorithm, the convolutional codes are conjugated to Reed–Solomon (RS) codes and RLL codes, which make use of soft information from the channel to provide coding gain. A modified Bahl, Cocke, Jelinek and Raviv algorithm is proposed to enhance convolutional decoding with the soft RLL decoder to produce soft input for RS decoder. Simulation results show that the bit error rate performance of our proposed algorithm is better than referenced algorithm.

Fast degree-distribution optimization for BATS codes

Batched sparse (BATS) codes have been proposed for communication through networks with packet loss. BATS codes include a matrix generalization of fountain codes as the outer code and random linear network coding at the intermediate network nodes as the inner code. BATS codes, however, do not possess a universal degree distribution that achieves an optimal rate for any distribution of the transfer matrix ranks. Therefore, it is important to have a fast degree-distribution optimization approach for finite-length BATS codes. In this paper, we propose the concept of batch release probability (BRP), and demonstrate some characteristics of BRPs from the degree distributions achieving nearly optimal performance. Based on these BRP characteristics, we propose a novel degree-distribution optimization approach that achieves the similar decoding performance with a much shorter optimization time, compared with the previous approach. Moreover, the universality of BRPs observed in this paper can further simplify the degree-distribution optimization of BATS codes.

Low-Complexity Soft-Decision Concatenated LDGM-Staircase FEC for High-Bit-Rate Fiber-Optic Communication

A concatenated soft-decision forward error correction (FEC) scheme consisting of an inner low-density generator-matrix (LDGM) code and an outer staircase code is proposed. The soft-decision LDGM code is used for error reduction, while the majority of bit errors are corrected by the low-complexity hard-decision staircase code. Decoding complexity of the concatenated code is quantified by a score based on the number of edges in the LDGM code Tanner graph, the number of decoding iterations, and the number of staircase code decoding operations. The inner LDGM ensemble is designed by solving an optimization problem, which minimizes the product of the average node degree and an estimate of the required number of decoding iterations. A search procedure is used to find the inner and outer code pair with lowest complexity. The design procedure results in a Pareto-frontier characterization of the tradeoff between net coding gain and complexity for the concatenated code. Simulations of code designs at $\text{20}\%$ overhead showed that the proposed scheme achieves net coding gains equivalent to existing soft-decision FEC solutions, with up to $\text{57}\%$ reduction in complexity.

Semi-Analytical Evaluation of Concatenated RS/LDPC Coding Performance With Finite Block Interleaving

The Monte Carlo (MC) simulation of the error performance of a concatenated coding system with finite interleaving depth between the outer and inner codes is time-consuming, especially when targeting low error rates and examining several interleaver settings. In this letter, we present a semi-analytical evaluation of the word error rate (WER) performance of a system with Reed–Solomon (RS) outer coding, finite block interleaving and systematic low-density parity check (LDPC) inner coding. The proposed evaluation method relies on a simple semi-analytical statistical model for the number of bit errors in a segment of the information word after LDPC decoding on the AWGN channel. Only the WER and bit error rate of the inner subsystem (LDPC code and the considered constellation) are required to compute the WER of the concatenated code, corresponding to different parameters of the RS code and the interleaver. We show that the semi-analytical WER of the concatenated system closely matches the WER resulting from MC simulations for both the AWGN and Rayleigh block-fading channels.

On the Error Performance of Coding and Equalization in Low-Complexity Ultra-Wideband Communication Systems

In this paper, the performance of various channel coding schemes is investigated in pulse-based ultra-wideband(UWB) communication systems for applications in short-range indoor environments. Pulse-based binary (BPSK) modulation and decision-feedback equalization (DFE) is considered. Concatenated adaptive equalization and coding is explored as an alternative to the more complex and often impractical joint coding and equalization. A block length of approximately 1000 bits is considered in this paper as it results in a static channel with minimal latency while still yielding relatively good error performance. The error performance of a previously proposed turbo product code (TPC), based on two identical Hamming (31,26) codes, is simulated and compared with that of otherchannel coding schemes of similar rate and code length. These include a regular LDPC (1057,813) code, a memory-6 rate-3/4 punctured convolutional code, a Reed-Solomon (127,89) code and a concatenated (off-the-shelf) code with a Reed-Solomon (255,239) outer code and a memory-6 rate-3/4 punctured convolutional inner code. The inclusion of the concatenated Reed-Solomon scheme serves as a reference, as this is an off-the-shelf classical and still popular solution. The simulation results show that, among the coding schemes considered, the LDPC code offers the best error performance.

Design of Generalized Concatenated Codes Based on Polar Codes With Very Short Outer Codes

We present a new design approach to construct generalized concatenated codes (GCCs) based on polar codes (PCs). It is already known that PCs can be considered as a special class of the GCCs with polar outer and inner codes. We show how density evolution (DE) can be used to develop a channel-specific method to design outer codes of length $L$ under maximum-likelihood (ML) decoding. Once a bank of outer codes of length $L$ and different rates have been designed, we develop a rate-allocation algorithm to allocate rates to the outer codes of the GCC with the goal of minimizing the overall block error rate for a given overall rate of $K/N$ . To maintain the low complexity of the SC decoder, we only design very short outer codes $(L\le\text{8})$ . We show that, at this outer code length, it is possible to design GCC-PCs that can result in up to 0.5-dB coding gain while reducing decoding complexity.

Concatenation of convolutional codes and rank metric codes for multi-shot network coding

In this paper we present a novel coding approach to deal with the transmission of information over a network. In particular we make use of the network several times (multi-shot) and impose correlation in the information symbols over time. We propose to encode the information via an inner and an outer code, namely, a Hamming metric convolutional code as an outer code and a rank metric code as an inner code. We show how this simple concatenation scheme can exploit the potential of both codes to produce a code that can correct a large number of error patterns.

Broadcast Coded Modulation: Multilevel and Bit-Interleaved Construction

The capacity of the AWGN broadcast channel is achieved by superposition coding, but the superposition of individual coded modulations expands the modulation alphabet and distorts its configuration. Hierarchical modulations, which appear in the literature mostly in the context of unequal error protection, can approach only a few isolated points on the boundary of the broadcast capacity region. This paper studies multilevel coding (MLC) for constellation-constrained broadcast-coded modulation. The conditions under which multilevel codes can achieve the constellation-constrained capacity of the AWGN broadcast channel are derived. We propose a pragmatic multilevel design technique with near-constellation-constrained-capacity performance where the coupling of the superposition inner and outer codes is localized to each bit-level. It is shown that this can be further relaxed to a code coupling on only one bit-level, with a little or no penalty under natural labeling. The rate allocation problem between the bit levels of the two users is studied and a pragmatic method is proposed, again with near-capacity performance. In further pursuit of lower complexity, a hybrid MLC-bit-interleaved-coded modulation is proposed, whose performance is shown to be very close to the boundary of the constellation-constrained capacity region. Simulation results show that good point-to-point LDPC codes produce excellent performance in the proposed coded modulation framework.

An Improved Concatenation Scheme of Polar Codes With Reed–Solomon Codes

An improved Reed–Solomon (RS)-polar code concatenation scheme with a threshold is proposed. It applies successive cancellation list and hard-decision decoding based on the low-complexity chase algorithm. A new multiplicity assignment module is proposed to make it possible to realize the RS-polar code concatenation and facilitate hardware decoder design. The proposed scheme achieves high decoding performance, low decoding complexity, and short latency. The simulation results show that the improved concatenation scheme gives a coding gain reaching about 0.4 dB compared with the traditional RS-polar scheme. Moreover, it can reduce the average number of RS codewords to be decoded in a concatenated code by 65% and decrease the decoding latency by 50% when the signal-to-noise ratio is 1.75 dB.

A General Construction for PMDS Codes

Partial MDS [(PMDS) also known as maximally recoverable] codes allow for local erasure recovery by utilizing row-wise parities and additional erasure correction through global parities. Recent works on PMDS codes focus on special case parameter settings, and a general construction for PMDS codes is stated as an open problem. This letter provides an explicit construction for PMDS codes for all parameters utilizing concatenation of Gabidulin and MDS codes, a technique originally proposed by Rawat et al. for constructing optimal locally repairable codes. This approach allows for PMDS constructions for any parameters albeit with large field sizes. To lower the field size, a relaxation on the rate requirement is considered, and PMDS codes based on combinatorial designs are constructed.

A Polynomial Formula for Finite-Length BATS Code Performance

Batched sparse (BATS) codes have been proposed for communication through networks with packet loss. BATS codes include a matrix generalization of fountain codes as the outer code and random linear network coding at the intermediate network nodes as the inner code. BATS codes, however, do not possess a universal degree distribution that achieves the optimal rate for any distribution of the transfer matrix ranks, so that fast performance evaluation of finite-length BATS codes is important for optimizing the degree distribution. The state-of-the-art finite-length performance evaluation method has a computational complexity of $\mathcal {O}(K^{2}n^{2}M)$ , where $K$ , $n$ , and $M$ are the number of input symbols, the number of batches, and the batch size, respectively. We propose a polynomial-form formula for finite-length BATS codes performance evaluation with the computational complexity of $\mathcal {O}(K^{2}n\ln n)$ . Numerical results demonstrate that the polynomial-form formula can be significantly faster than the previous methods.

Multipartite entangled states, symmetric matrices and error-correcting codes

A pure quantum state is called $k$ -uniform if all its reductions to $k$ -qudit are maximally mixed. We investigate the general constructions of $k$ -uniform pure quantum states of $n$ subsystems with $d$ levels. We provide one construction via symmetric matrices and the second one through the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even $n\ge 2$ , there always exists an $n/2$ -uniform $n$ -qudit quantum state of level $p$ for sufficiently large prime $p$ . Second, both constructions show that there exist $k$ -uniform $n$ -qudit pure quantum states such that $k$ is proportional to $n$ , i.e., $k=\Omega (n)$ although the construction from symmetric matrices in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform $n$ -qudit pure quantum state for all $n\ge 8$ . In fact, we can further prove that, for every $k$ , there exists a constant $M_{k}$ such that there exists a $k$ -uniform $n$ -qudit quantum state for all $n\ge M_{k}$ . In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of $k$ -uniform quantum state when $k$ tends to infinity.

Low Error Floor Concatenated LDPC for MIMO Systems

Multiple-Input and Multiple-Output, or MIMO is the use of multiple antennas at both the transmitter and receiver to improve communication performance. MIMO technology has attracted attention in wireless communications; because it offers significant increases in data throughput and spectral efficiency without additional bandwidth or increased transmit power. To achieve the mentioned above performance Bit Error Rates (BER) should be low. For this reason efficient encoding and decoding algorithms should be used. MIMO systems rely on error-control coding to ensure reliable communication in the presence of noise. Forward Error Correction Codes (FEC) such as convolutional and block codes were investigated for MIMO systems. Low Density Parity Check (LDPC) shows good performance except that an error floor may appear at high Signal to Noise Ratio (SNR). In this work we propose a concatenated error control code that reduces the error floor of LDPC codes suffering from error floor. The proposed scheme is a good candidate for high rates real time communication since it reduces the decoding latency as well.