Decoding Golay Codes and their Related Lattices: A PAC Code Perspective
College of Computer Science and Software Engineering, Shenzhen Univeristy（深圳大学计算机科学与软件工程学院）；Guangzhou Institute of Technology, Xidian University（西安电子科技大学广州研究院）；College of Cyber Security, Jinan University（暨南大学网络空间安全学院）
Yujun Ji, Ling Liu, Shanxiang Lyu, Chao Chen, Tao Dai, Baoming Bai
In this work, we propose a decoding method of Golay codes from the perspective of Polarization Adjusted Convolutional (PAC) codes. By invoking Forney's cubing construction of Golay codes and their generators $G^*(8,7)/(8,4)$, we found different construction methods of Golay codes from PAC codes, which result in an efficient parallel list decoding algorithm with near-maximum likelihood performance. Compared with existing methods, our method can get rid of index permutation and codeword puncturing. Using the new decoding method, some related lattices, such as Leech lattice $Λ_{24}$ and its principal sublattice $H_{24}$, can be also decoded efficiently.
We show that the binary Golay code is slightly more power efficient than the extended binary Golay code under maximum-likelihood (ML), hard-decision decoding. In fact, if a codeword from the extended code is transmitted, one cannot achieve a higher probability of correct decoding than by simply ignoring the 24th symbol and using an ML decoder for the non-extended code on the first 23 symbols. This is so, despite the fact that using that last symbol would allow one to sometimes correct error patterns with weight four. To our knowledge the worse performance of the extended Golay code has not bee…
The extended Golay code is shown to be representable as a chained polar subcode. This enables its decoding with the successive cancellation algorithm and its stack generalization. The decoder can be further simplified by employing fast Hadamard transform. The complexity of the obtained algorithm is comparable with that of the Vardy algorithm.
We propose two approximate algorithms for MAP decoding on tail-biting trellises. The algorithms work on a subset of nodes of the tail-biting trellis, judiciously selected. We report the results of simulations on an AWGN channel using the approximate algorithms on tail-biting trellises for the $(24,12)$ Extended Golay Code and a rate 1/2 convolutional code with memory 6.
In this paper we explore the decision regions of Linear Programming (LP) decoding. We compare the decision regions of an LP decoder, a Belief Propagation (BP) decoder and the optimal Maximum Likelihood (ML) decoder. We study the effect of minimal-weight pseudocodewords on LP decoding. We present global optimization as a method for finding the minimal pseudoweight of a given code as well as the number of minimal-weight generators. We present a complete pseudoweight distribution for the [24; 12; 8] extended Golay code, and provide justifications of why the pseudoweight distribution alone cannot …
We develop a framework for linear-programming (LP) decoding of non-binary linear codes over rings. We prove that the resulting LP decoder has the `maximum likelihood certificate' property, and we show that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. LP decoding performance is illustrated for the (11,6,5) ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the LP decoder performance is comparable to codeword-error-rate-optimum hard-decision b…
A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the 'maximum-likelihood certificate' property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particul…
Performance and complexity of sequential decoding of polarization-adjusted convolutional (PAC) codes is studied. In particular, a performance and computational complexity comparison of PAC codes with 5G polar codes and convolutional codes is given. A method for bounding the complexity of sequential decoding of PAC codes is proposed.
Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short block lengths is far from optimal. Arikan has recently presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. Such PAC codes provide dramatic improvement in performance as compared to both standard successive-cancellation decoding as well as CRC-aided list decoding. Arikan's PAC codes are based primarily upon the following ideas: replacing CRC precoding wit…
A latest coding scheme named polarization-adjusted convolutional (PAC) codes is shown to approach the dispersion bound for the code (128,64) under list decoding. However, to achieve the near-bound performance, the list size of list decoding needs to be excessively large, which leads to insufferable latency. In this paper, to improve the speed of list decoding, fast list decoders for PAC codes are proposed. We define four types of constituent nodes and provide fast list decoding algorithms for each of them. Simulation results present that fast list decoding with three types of constituent nodes…
Despite the extreme error-correction performance, the amount of computation of sequential decoding of the polarization-adjusted convolutional (PAC) codes is random. In sequential decoding of convolutional codes, the computational cutoff rate denotes the region between rates whose average computational complexity of decoding is finite and those which is infinite. In this paper, by benefiting from the polarization and guessing techniques, we prove that the computational cutoff rate in sequential decoding of pre-transformed polar codes polarizes. The polarization of the computational cutoff rate …
In this paper, we propose to enhance the performance of the sum-product algorithm (SPA) by interleaving SPA iterations with a random local graph update rule. This rule is known as edge local complementation (ELC), and has the effect of modifying the Tanner graph while preserving the code. We have previously shown how the ELC operation can be used to implement an iterative permutation group decoder (SPA-PD)--one of the most successful iterative soft-decision decoding strategies at small blocklengths. In this work, we exploit the fact that ELC can also give structurally distinct parity-check mat…
Practical random network coding based schemes for multicast include a header in each packet that records the transformation between the sources and the terminal. The header introduces an overhead that can be significant in certain scenarios. In previous work, parity check matrices of error control codes along with error decoding were used to reduce this overhead. In this work we propose novel packet formats that allow us to use erasure decoding and list decoding. Both schemes have a smaller overhead compared to the error decoding based scheme, when the number of sources combined in a packet is…
Convolutional precoding in polarization-adjusted convolutional (PAC) codes can reduce the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This can result in improving the error correction performance of (near) maximum likelihood (ML) decoders such as sequential decoders and sphere decoders. However, PAC codes cannot be decoded by sphere decoding. The reason is twofold: 1) Sphere decoding of polar codes is performed from the last bit - due to the lower rectangular shape of the polar transform. Whereas the shape of PAC codes generator matrix is no longer triangular. …
Polarization-adjusted convolutional (PAC) codes are special concatenated codes in which we employ a one-to-one convolutional transform as a pre-coding step before the polar transform. In this scheme, the polar transform (as a mapper) and the successive cancellation process (as a demapper) present a synthetic vector channel to the convolutional transformation. The numerical results show that this concatenation improves the Hamming distance properties of polar codes. In this work, we implement the parallel list Viterbi algorithm (LVA) and show how the error correction performance moves from the …
In this paper, we present an optimal metric function on average, which leads to a significantly low decoding computation while maintaining the superiority of the polarization-adjusted convolutional (PAC) codes' error-correction performance. With our proposed metric function, the PAC codes' decoding computation is comparable to the conventional convolutional codes (CC) sequential decoding. Moreover, simulation results show an improvement in the low-rate PAC codes' error-correction performance when using our proposed metric function. We prove that choosing the polarized cutoff rate as the metric…
This paper proposes a rate-profile construction method for polarization-adjusted convolutional (PAC) codes of any code length and rate, which is capable of maintaining trade-off between the error-correction performance and decoding complexity of PAC code. The proposed method can improve the error-correction performance of PAC codes while guaranteeing a low mean sequential decoding complexity for signal-to-noise ratio (SNR) values beyond a target SNR value.
Raptor codes have been widely used in many multimedia broadcast/multicast applications. However, our understanding of Raptor codes is still incomplete due to the insufficient amount of theoretical work on the performance analysis of Raptor codes, particularly under maximum-likelihood (ML) decoding, which provides an optimal benchmark on the system performance for the other decoding schemes to compare against. For the first time, this paper provides an upper bound and a lower bound, on the packet error performance of Raptor codes under ML decoding, which is measured by the probability that all …
Sorting operation is one of the main bottlenecks for the successive-cancellation list (SCL) decoding. This paper introduces an improvement to the SCL decoding for polar and pre-transformed polar codes that reduces the number of sorting operations without degrading the code's error-correction performance. In an SCL decoding with an optimum metric function we show that, on average, the correct branch's bit-metric value must be equal to the bit-channel capacity, and on the other hand, the average bit-metric value of a wrong branch can be at most zero. This implies that a wrong path's partial path…
Fountain codes are erasure correcting codes realizing reliable communication systems for the multicast on the Internet. The zigzag decodable fountain (ZDF) code is one of generalization of the Raptor code, i.e, applying shift operation to generate the output packets. The ZDF code is decoded by a two-stage iterative decoding algorithm, which combines the packet-wise peeling algorithm and the bit-wise peeling algorithm. By the bit-wise peeling algorithm and shift operation, ZDF codes outperform Raptor codes under iterative decoding in terms of decoding erasure rate and overhead. However, the bit…
Random linear network coding (RLNC) in theory achieves the max-flow capacity of multicast networks, at the cost of high decoding complexity. To improve the performance-complexity tradeoff, we consider the design of sparse network codes. A generation-based strategy is employed in which source packets are grouped into overlapping subsets called generations. RLNC is performed only amongst packets belonging to the same generation throughout the network so that sparseness can be maintained. In this paper, generation-based network codes with low reception overheads and decoding costs are designed fo…
We consider binary systematic network codes and investigate their capability of decoding a source message either in full or in part. We carry out a probability analysis, derive closed-form expressions for the decoding probability and show that systematic network coding outperforms conventional network coding. We also develop an algorithm based on Gaussian elimination that allows progressive decoding of source packets. Simulation results show that the proposed decoding algorithm can achieve the theoretical optimal performance. Furthermore, we demonstrate that systematic network codes equipped w…
Optimization of Sparse VLSF Codes for Short-Packet Transmission via Saddlepoint Methods
Centre Inria d’Université Côte d’Azur, Inria（蔚蓝海岸大学Inria中心）；Laboratoire GAATI, Université de la Polynésie française（法属波利尼西亚大学GAATI实验室）；ECE Dept, Princeton University（普林斯顿大学电气与计算机工程系）；Univ. Rennes, INSA, CNRS, IETR UMR 6164（雷恩大学、国立应用科学学院、法国国家科学研究中心IETR联合实验室）；Inria, INSA Lyon, CITI Laboratory（里昂国立应用科学学院CITI实验室）
Simulation results illustrating the performance and complexity of the sequential successive cancellation decoding algorithm are presented for the case of polar subcodes with Arikan and large kernels, as well as for extended BCH\ codes. Performance comparison with Arikan PAC and LDPC codes is provided. Furthermore, complete description of the decoding algorithm is presented.
This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay $T_\mathrm{v}$ and every source message in the second sequence subject to a shorter decoding delay $T_\mathrm{u}\le T_\mathrm{v}$. We assume that the channel loss mo…
This paper studies the decoding capabilities of maximum distance profile (MDP) convolutional codes over the erasure channel and compares them with the decoding capabilities of MDS block codes over the same channel. The erasure channel involving large alphabets is an important practical channel model when studying packet transmissions over a network, e.g, the Internet.
The polarization-adjusted convolutional (PAC) codes concatenate the polar transform and the convolutional transform to improve the decoding performance of the finite-length polar codes, where the rate-profile is used to construct the PAC codes by setting the positions of frozen bits. However, the optimal rateprofile method of PAC codes is still unknown. In this paper, an optimized rate-profile algorithm of PAC codes is proposed. First, we propose the normalized compression factor (NCF) to quantify the transmission efficiency of useful information, showing that the distribution of useful inform…
Recent work have shown that Reed-Muller (RM) codes achieve the erasure channel capacity. However, this performance is obtained with maximum-likelihood decoding which can be costly for practical applications. In this paper, we propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure channel based on Plotkin construction. We present several improvements over the generic decoding. They allow, for a light cost, to compete with maximum-likelihood decoding performance, especially on high-rate codes, while significantly outperforming it in terms of speed.
Error control is significant to network coding, since when unchecked, errors greatly deteriorate the throughput gains of network coding and seriously undermine both reliability and security of data. Two families of codes, subspace and rank metric codes, have been used to provide error control for random linear network coding. In this paper, we enhance the error correction capability of these two families of codes by using a novel two-tier decoding scheme. While the decoding of subspace and rank metric codes serves a second-tier decoding, we propose to perform a first-tier decoding on the packe…
This paper proposes a fountain coding system which has lower space decoding complexity and lower decoding erasure rate than the Raptor coding systems. The main idea of the proposed fountain code is employing shift and exclusive OR to generate the output packets. This technique is known as the zigzag decodable code, which is efficiently decoded by the zigzag decoder. In other words, we propose a fountain code based on the zigzag decodable code in this paper. Moreover, we analyze the overhead for the received packets, decoding erasure rate, decoding complexity, and asymptotic overhead of the pro…
Several types of AL-FEC (Application-Level FEC) codes for the Packet Erasure Channel exist. Random Linear Codes (RLC), where redundancy packets consist of random linear combinations of source packets over a certain finite field, are a simple yet efficient coding technique, for instance massively used for Network Coding applications. However the price to pay is a high encoding and decoding complexity, especially when working on $GF(2^8)$, which seriously limits the number of packets in the encoding window. On the opposite, structured block codes have been designed for situations where the set o…
Sun Yat-sen University（中山大学）；Guangdong Province Key Laboratory of Information Security Technology（广东省信息安全技术重点实验室）；Huawei Technologies Co., Ltd.（华为技术有限公司）
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributio…
This paper deals with two main issues regarding the short polar codes: the potential of FEC-assisted decoding and optimal code concatenation strategies under various design scenarios. Code concatenation and FEC-assisted decoding are presented systematically, assuming a packetized system. It is shown that FEC-assisted decoding can improve frame error rate of successive cancellation decoding arbitrarily, at the expense of some coding rate loss and decoding complexity linearly increasing with the number of codewords in the frame. This is compared with list decoding whose complexity grows linearly…
We study some structural properties of Construction-A lattices obtained from low density parity check (LDPC) codes over prime fields. Such lattices are called low density Construction-A (LDA) lattices, and permit low-complexity belief propagation decoding for transmission over Gaussian channels. It has been shown that LDA lattices achieve the capacity of the power constrained additive white Gaussian noise (AWGN) channel with closest lattice-point decoding, and simulations suggested that they perform well under belief propagation decoding. We continue this line of work, and prove that these lat…
We revisit the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set $\mathcal{C}$ of points in $ \mathbb{R}^n $ such that any point in $ \mathbb{R}^n $ lies in the intersection of at most $ L-1 $ balls of radius $ \sqrt{nN} $ around points in $ \mathcal{C} $. We study the multiple packing problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose poi…
Recently, Martinez-Penas and Kschischang (IEEE Trans. Inf. Theory, 2019) showed that lifted linearized Reed-Solomon codes are suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed-Solomon codes. Compared to the construction by Martinez-Penas-Kschischang, interleaving allows to increase the decoding region significantly (especially w.r.t. the number of insertions) and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. The proposed decoder is a list deco…
In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$. One needs to recover any erased packet, where erasures are as per the channel model, with a strict decoding delay deadline of $τ$ time slots. Presently existing rate-optimal constructions in the literature require, in general, a field-size which gro…
We present constructions of Space-Time (ST) codes based on lattice coset coding. First, we focus on ST code constructions for the short block-length case, i.e., when the block-length is equal to or slightly larger than the number of transmit antennas. We present constructions based on dense lattice packings and nested lattice (Voronoi) shaping. Our codes achieve the optimal diversity-multiplexing tradeoff of quasi-static MIMO fading channels for any fading statistics, and perform very well also at practical, moderate values of signal to noise ratios (SNR). Then, we extend the construction to t…
In this paper, network error control coding is studied for robust and efficient multicast in a directed acyclic network with imperfect links. The block network error control coding framework, BNEC, is presented and the capability of the scheme to correct a mixture of symbol errors and packet erasures and to detect symbol errors is studied. The idea of syndrome-based decoding and error detection is introduced for BNEC, which removes the effect of input data and hence decreases the complexity. Next, an efficient three-stage syndrome-based BNEC decoding scheme for network error correction is prop…
This paper reviews the potential channel decoding techniques for ultra-reliable low-latency communications (URLLC). URLLC is renowned for its stringent requirements including ultra-reliability, low end-to-end transmission latency, and packet-size flexibility. These requirements exacerbate the difficulty of the physical-layer design, particularly for the channel coding and decoding schemes. To satisfy the requirements of URLLC, decoders must exhibit superior error-rate performance \black{and} low decoding complexity. \black{Also, it is desired that decoders be universal} to accommodate various …
BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding. Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a give…
Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain vector space over a finite field. The Koetter-Kschischang construction of subspace codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang subspace codes. In a sense, we are able to achieve for these codes what Sudan was …
Recently, the sparse vector code (SVC) is emerging as a promising solution for short-packet transmission in massive machine type communication (mMTC) as well as ultra-reliable and low-latency communication (URLLC). In the SVC process, the encoding and decoding stages are jointly modeled as a standard compressed sensing (CS) problem. Hence, this paper aims at improving the decoding performance of SVC by optimizing the spreading matrix (i.e. measurement matrix in CS). To this end, two greedy algorithms to minimize the mutual coherence value of the spreading matrix in SVC are proposed. Specially,…
Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most $L$. For odd $L\ge 3$ an asymptotic upper bound on the rate of any such packing is proven. Resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. Method is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn (that was used previously to improve the estimates of Blinovsky for $L=2$) and a Ramsey-theoretic technique of Blinovsky. As an applica…
This brief proposes a hardware implementation architecture for Fano decoding of polarization-adjusted convolutional (PAC) codes. This architecture uses a novel branch metric unit specific to PAC codes. The proposed decoder is tested on FPGA, and its performance is evaluated on ASIC using TSMC 28 nm 0.72 V library. The decoder can be clocked at 500 MHz and reach an average information throughput of 38 Mb/s at 3.5 dB signal-to-noise ratio for a block length of 128 and a code rate of 1/2.
