Showing papers on "List decoding published in 2020"

AI Coding: Learning to Construct Error Correction Codes

Abstract: In this paper, we investigate an artificial-intelligence (AI) driven approach to design error correction codes (ECC). Classic error-correction code design based upon coding-theoretic principles typically strives to optimize some performance-related code property such as minimum Hamming distance, decoding threshold, or subchannel reliability ordering. In contrast, AI-driven approaches, such as reinforcement learning (RL) and genetic algorithms, rely primarily on optimization methods to learn the parameters of an optimal code within a certain code family. We employ a constructor-evaluator framework, in which the code constructor can be realized by various AI algorithms and the code evaluator provides code performance metric measurements. The code constructor keeps improving the code construction to maximize code performance that is evaluated by the code evaluator. As examples, we focus on RL and genetic algorithms to construct linear block codes and polar codes. The results show that comparable code performance can be achieved with respect to the existing codes. It is noteworthy that our method can provide superior performances to classic constructions in certain cases (e.g., list decoding for polar codes).

CRC-Aided Belief Propagation List Decoding of Polar Codes

Abstract: Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopping condition (i.e., for error-detection), we also aim to benefit from the error-correction capabilities of the outer CRC code. For this, we develop two distinct soft-decision CRC decoding algorithms: a Bahl-Cocke-Jelinek-Raviv (BCJR)-based approach and a sum product algorithm (SPA)-based approach. Further, an optimized selection of permuted factor graphs is analyzed and shown to reduce the decoding complexity significantly. Finally, we benchmark the proposed CRC-aided belief propagation list (CA-BPL) decoding to state-of-the-art 5G polar codes under CA-SCL decoding and, thereby, showcase an error-rate performance not just close to the CA-SCL but also close to the maximum likelihood (ML) bound as estimated by ordered statistic decoding (OSD).

Optimally resilient codes for list-decoding from insertions and deletions

Abstract: We give a complete answer to the following basic question: ”What is the maximal fraction of deletions or insertions tolerable by q-ary list-decodable codes with non-vanishing information rate?” This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best-known result was that a γ≤ 0.707 fraction of insertions is tolerable by some binary code family. For any desired є>0, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of γ fraction of insertions and δ fraction of deletions as long as γ+2δ≤ 1−e. On the other hand, for any γ, δ with γ+2δ=1 list-decoding is impossible. Our result thus precisely characterizes the feasibility region of binary list-decodable codes for insertions and deletions. We further generalize our result to codes over any finite alphabet of size q. Surprisingly, our work reveals that the feasibility region for q>2 is not the natural generalization of the binary bound above. We provide tight upper and lower bounds that precisely pin down the feasibility region, which turns out to have a (q−1)-piece-wise linear boundary whose q corner-points lie on a quadratic curve. The main technical work in our results is proving the existence of code families of sufficiently large size with good list-decoding properties for any combination of δ,γ within the claimed feasibility region. We achieve this via an intricate analysis of codes introduced by [Bukh, Ma; SIAM J. Discrete Math; 2014]. Finally, we give a simple yet powerful concatenation scheme for list-decodable insertion-deletion codes which transforms any such (non-efficient) code family (with information rate zero) into an efficiently decodable code family with constant rate.

List Decoding of Arıkan’s PAC Codes

Abstract: 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 under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short block lengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has remained the state of the art in polar coding since it was first introduced in 2011. Last year, Arikan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. Such PAC codes provide another dramatic improvement in performance as compared to CRC-aided list decoding. These codes are based primarily upon the following main ideas: replacing CRC precoding with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. Arikan’s simulation results show that PAC codes, resulting from the combination of these ideas, are quite close to finite-length lower bounds on the performance of any code under ML decoding.One of our main goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, L ⩾ 128). List decoding has distinct advantages over sequential decoding in certain scenarios such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance under ML decoding. These results indicate that PAC codes are superior to polar codes and Reed-Muller codes, and suggest that the goal of rate-profiling may be to optimize the weight distribution at low weights.

An algorithm for decoding skew Reed–Solomon codes with respect to the skew metric

Abstract: After giving a new interpretation of the skew metric defined in [8], we show that the decoding algorithm of [4] for skew Reed–Solomon codes in the Hamming metric remains valid with respect to the skew metric. This enables us to make a first step towards a list decoding algorithm in the skew metric.

LDPC Codes Achieve List Decoding Capacity

Abstract: We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieves list-decoding capacity with high probability. These are the first graph-based codes shown to have this property. This result opens up a potential avenue towards truly linear-time list-decodable codes that achieve list-decoding capacity. Our result on list decoding follows from a much more general result: any local property satisfied with high probability by a random linear code is also satisfied with high probability by a random LDPC code from Gallager's distribution. Local properties are properties characterized by the exclusion of small sets of codewords, and include list-decoding, list-recovery and average-radius list-decoding. In order to prove our results on LDPC codes, we establish sharp thresholds for when local properties are satisfied by a random linear code. More precisely, we show that for any local property $\mathcal{P}$ , there is some $R^{\ast}$ so that random linear codes of rate slightly less than $R^{\ast}$ satisfy $\mathcal{P}$ with high probability, while random linear codes of rate slightly more than $R^{\ast}$ with high probability do not. We also give a characterization of the threshold rate $R^{\ast}$ . This is an extended abstract. The full version is available at https://arxiv.org/abs/1909.06430

List Decodable Mean Estimation in Nearly Linear Time

Abstract: Learning from data in the presence of outliers is a fundamental problem in statistics. Until recently, no computationally efficient algorithms were known to compute the mean of a high dimensional distribution under natural assumptions in the presence of even a small fraction of outliers. In this paper, we consider robust statistics in the presence of overwhelming outliers where the majority of the dataset is introduced adversarially. With only an $\alpha < 1/2$ fraction of "inliers" (clean data) the mean of a distribution is unidentifiable. However, in their influential work, [CSV17] introduces a polynomial time algorithm recovering the mean of distributions with bounded covariance by outputting a succinct list of $O(1/\alpha)$ candidate solutions, one of which is guaranteed to be close to the true distributional mean; a direct analog of 'List Decoding' in the theory of error correcting codes. In this work, we develop an algorithm for list decodable mean estimation in the same setting achieving up to constants the information theoretically optimal recovery, optimal sample complexity, and in nearly linear time up to polylogarithmic factors in dimension. Our conceptual innovation is to design a descent style algorithm on a nonconvex landscape, iteratively removing minima to generate a succinct list of solutions. Our runtime bottleneck is a saddle-point optimization for which we design custom primal dual solvers for generalized packing and covering SDP's under Ky-Fan norms, which may be of independent interest.

Polarization-adjusted Convolutional (PAC) Codes: Fano Decoding vs List Decoding.

Abstract: In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arikan proposed to employ a one-to-one convolutional transform as a pre-coding step before polar transform. The resulting codes of this concatenation are called {\em polarization-adjusted convolutional (PAC) codes}. In this scheme, a pair of polar mapper and demapper as pre- and post-processing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of outer code. In this paper, the implementations of list decoding and Fano decoding for PAC codes are first investigated. Then, in order to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to reducing the average decoding time complexity up to 85\%, with only a relatively small degradation in error correction performance. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is necessary for backtracking and avoids storing the intermediate information or restarting the decoding process.

List Decodable Subspace Recovery

Abstract: Learning from data in the presence of outliers is a fundamental problem in statistics. In this work, we study robust statistics in the presence of overwhelming outliers for the fundamental problem of subspace recovery. Given a dataset where an $\alpha$ fraction (less than half) of the data is distributed uniformly in an unknown $k$ dimensional subspace in $d$ dimensions, and with no additional assumptions on the remaining data, the goal is to recover a succinct list of $O(\frac{1}{\alpha})$ subspaces one of which is nontrivially correlated with the planted subspace. We provide the first polynomial time algorithm for the 'list decodable subspace recovery' problem, and subsume it under a more general framework of list decoding over distributions that are "certifiably resilient" capturing state of the art results for list decodable mean estimation and regression.

List decoding of direct sum codes

Abstract: We consider families of codes obtained by "lifting" a base code C through operations such as k-XOR applied to "local views" of codewords of C, according to a suitable k-uniform hypergraph. The k-XOR operation yields the direct sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are indeed satisfied by the collection of length k walks on a sufficiently strong expanding graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings corresponding to the above hypergraph families. Using known connections between direct sum and direct product, we also recover (and strengthen) the recent results of Dinur et al. [SODA 2019] on list decoding for direct product liftings. Our framework relies on relaxations given by the Sum-of-Squares (SOS) SDP hierarchy for solving various constraint satisfaction problems (CSPs). We view the problem of recovering the closest codeword to a given (possibly corrupted) word, as finding the optimal solution to an instance of a CSP. Constraints in the instance correspond to edges of the lifting hypergraph, and the solutions are restricted to lie in the base code C. We show that recent algorithms for (approximately) solving CSPs on certain expanding hypergraphs by some of the authors also yield a decoding algorithm for such lifted codes. We extend the framework to list decoding, by requiring the SOS solution to minimize a convex proxy for negative entropy. We show that this ensures a covering property for the SOS solution, and the "condition and round" approach used in several SOS algorithms can then be used to recover the required list of codewords.

Generalized List Decoding

Abstract: This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR) channels, erasure channels, AND (Z-) channels, OR (ℤ-) channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (L − 1)-list decodable codes (where the list size L is a universal constant) exist for such channels. Our criterion essentially asserts that:For any given general adversarial channel, it is possible to construct positive rate (L − 1)-list decodable codes if and only if the set of completely positive tensors of order-L with admissible marginals is not entirely contained in the order-L confusability set associated to the channel.The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by1. extracting approximately equicoupled subcodes (generalization of equidistant codes) from any sequence of "large" codes using hypergraph Ramsey’s theorem, and2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone.In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which may be of independent interest.Other results include1 List decoding capacity with asymptotically large L for general adversarial channels;2 A tight list size bound for most constant composition codes (generalization of constant weight codes);3 Rederivation and demystification of Blinovsky’s [9] characterization of the list decoding Plotkin points (threshold at which large codes are impossible) for bit-flip channels;4 Evaluation of general bounds ([43]) for unique decoding in the error correction code setting.

List Viterbi Decoding of PAC Codes

Abstract: 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 poor performance of the Viterbi algorithm (VA) to the superior performance of list decoding by changing the constraint length, list size, and the sorting strategy (local sorting and global sorting) in the LVA. Also, we analyze the latency of the local sorting of the paths in LVA relative to the global sorting in the list decoding and the trade-off between the sorting latency and the error correction performance.

A 506Gbit/s Polar Successive Cancellation List Decoder with CRC

Abstract: Polar codes have recently attracted significant attention due to their excellent error-correction capabilities. However, efficient decoding of Polar codes for high throughput is very challenging. Beyond 5G, data rates towards 1Tbit/s are expected. Low complexity decoding algorithms like Successive Cancellation (SC) decoding enable such high throughput but suffer on errorcorrection performance. Polar Successive Cancellation List (SCL) decoders, with and without Cyclic Redundancy Check (CRC), exhibit a much better error-correction but imply higher implementation cost. In this paper we in-depth investigate and quantify various trade-offs of these decoding algorithms with respect to error-correction capability and implementation costs in terms of area, throughput and energy efficiency in a 28nm CMOS FD-SOI technology. We present a framework that automatically generates decoder architectures for throughputs beyond 100Gbit/s. This framework includes various architectural optimizations for SCL decoders that go beyond State-of-the-Art. We demonstrate a 506Gbit/s SCL decoder with CRC that was generated by this framework.

Soft List Decoding of Polar Codes

Abstract: Soft-output (SO) decoding is proposed for the Logarithmic Successive Cancellation List (Log-SCL) polar decoder for the first time, by exploiting the left-to-right propagation of the Belief Propagation (BP) decoder, which opens new avenues for its employment in powerful turbo-receivers. In the case of decoding a half-rate polar code having a block length of 1024 bits, the proposed soft list polar decoder achieves a 1.5 dB Block Error Ratio (BLER) performance gain, $50\%$ latency improvement and $26\%$ complexity reduction, compared to the state-of-the-art SO Soft Cancellation (SCAN) polar decoder in a polar-coded Multiple-Input Multiple-Output (MIMO) system. Furthermore, we conceive a Memory-Efficient (ME) soft list polar decoder, which requires only $16\%$ of the soft list polar decoder's memory, at the cost of slightly increased latency and complexity.

SCAN List Decoding of Polar Codes.

Abstract: In this paper we propose an enhanced soft cancellation (SCAN) decoder for polar codes based on decoding stages permutation. The proposed soft cancellation list (SCANL) decoder runs L independent SCAN decoders, each one relying on a different permuted factor graph. The estimated bits are selected among the L candidates through a dedicated metric provided by the decoders. Furthermore, we introduce an early-termination scheme reducing decoding latency without affecting error correction performance. We investigate the error-correction performance of the proposed scheme under various combinations of number of iterations used, permutation set and early-termination condition. Simulation results show that the proposed SCANL provides similar results when compared with belief propagation list, while having a smaller complexity. Moreover, for large list sizes, SCANL outperforms non-CRC aided successive cancellation list decoding.

Multi-Kernel Polar Codes: Concept and Design Principles

Abstract: In this paper, we propose a new polar code construction by employing kernels of different sizes in the Kronecker product of the transformation matrix, thus generalizing the original construction by Arikan. The proposed multi-kernel polar code allows for more flexibility in terms of the code length, moreover allowing for various new design principles. We describe in detail encoding as well as successive cancellation (SC) decoding and SC list (SCL) decoding, and we provide a novel design method for the frozen set that allows to optimise the performance under list decoding, as opposed to original relability-based code design. Finally, we numerically demonstrate the advantage of multi-kernel polar codes under the new design principles compared to punctured and shortened polar codes.

Combinatorial list-decoding of Reed-Solomon codes beyond the Johnson radius

Abstract: List-decoding of Reed-Solomon (RS) codes beyond the so called Johnson radius has been one of the main open questions in coding theory and theoretical computer science since the work of Guruswami and Sudan. It is now known by the work of Rudra and Wootters, using techniques from high dimensional probability, that over large enough alphabets there exist RS codes that are indeed list-decodable beyond this radius. In this paper we take a more combinatorial approach that allows us to determine the precise relation (up to the exact constant) between the decoding radius and the list size. We prove a generalized Singleton bound for a given list size, and conjecture that the bound is tight for most RS codes over large enough finite fields. We also show that the conjecture holds true for list sizes 2 and 3, and as a by product show that most RS codes with a rate of at least 1/9 are list-decodable beyond the Johnson radius. Lastly, we give the first explicit construction of such RS codes. The main tools used in the proof are a new type of linear dependency between codewords of a code that are contained in a small Hamming ball, and the notion of cycle space from Graph Theory. Both of them have not been used before in the context of list-decoding.

Generalized List Decoding

Abstract: This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR) channels, erasure channels, AND (Z-) channels, OR channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (L-1)-list decodable codes (where the list size L is a universal constant) exist for such channels. Our criterion essentially asserts that: For any given general adversarial channel, it is possible to construct positive rate (L-1)-list decodable codes if and only if the set of completely positive tensors of order-L with admissible marginals is not entirely contained in the order-L confusability set associated to the channel. The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1) extracting approximately equicoupled subcodes (generalization of equidistant codes) from any using hypergraph Ramsey’s theorem, and 2) significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which may be of independent interest. Other results include 1) List decoding capacity with asymptotically large L for general adversarial channels; 2) A tight list size bound for most constant composition codes (generalization of constant weight codes); 3) Rederivation and demystification of Blinovsky’s [Blinovsky, 1986] characterization of the list decoding Plotkin points (threshold at which large codes are impossible) for bit-flip channels; 4) Evaluation of general bounds [Wang et al., 2019] for unique decoding in the error correction code setting.

The complexity of the functional security assessment algorithm for information technologies for the creation of warranty automated systems

Abstract: The complexity of the algorithm of communication of the system of linear levels with open regular parts by means of list decoding of "shortened" codes of reed painters which are intended for use in methods of an estimation of functional safety of cryptographic algorithms of cryptographic subsystems of the guaranteed automated systems creating on objects of critical infrastructure and socially important objects. This paper proposes solving problems to assess the complexity of the proposed algorithm. As a result, the upper estimates of the average labor productivity for the general case and the maximum complexity of the proposed algorithm for many special reviews related to the restoration of the formed linear results of the maximum period over a field of two elements. The achievable upper part of the list, which is formed using the proposed algorithm, is also indicated. The obtained results indicate that with certain collaborations between the parameters of the previously proposed algorithm, the time complexity was changed in comparison with the previously known deterministic algorithm for a similar purpose, which is based on the fast Hadamara transformation. This means that a more effective tool can be used to assess the impact of cryptographic subsystems on powerful cyberattacks to obtain a more accurate assessment of their functional security.

Combinatorial Group Testing and Sparse Recovery Schemes with Near-Optimal Decoding Time

Abstract: In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$ , using $m\ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs with efficient decoding algorithms. The most efficient decoders should run in sublinear time in $n$ , and ideally near-linear in the number of measurements $m$ . In this work, we give several constructions with an optimal number of measurements and near-optimal decoding time for the most fundamental group testing tasks, as well as for central tasks in the compressed sensing and heavy hitters literature. For many of those tasks, the previous measurement-optimal constructions needed time either quadratic in the number of measurements or linear in the universe size. Among our results are the following: a construction of disjunct matrices matching the best-known construction in terms of the number of rows $m$ , but achieving nearly linear decoding time in $m$ ; a construction of list disjunct matrices with the optimal $m=O(k\log(n/k)$ number of rows and nearly linear decoding time in $m$ ; error-tolerant variations of the above constructions; a non-adaptive group testing scheme for the “for-each” model with $m=O(k\log n)$ measurements and $O(m)$ decoding time; a streaming algorithm for the “for-all” version of the heavy hitters problem in the strict turnstile model with near-optimal query time, as well as a “list decoding” variant obtaining also near-optimal update time and $O(k\log(n/k))$ space usage; an $\ell_{2}/\ell_{2}$ weak identification system for compressed sensing with nearly optimal sample complexity and nearly linear decoding time in the sketch length. Most of our results are obtained via a clean and novel approach that avoids list-recoverable codes or related complex techniques that were present in almost every state-of-the-art work on efficiently decodable constructions of such objects.

Non-Orthogonal Modulation for Short Packets in Massive Machine Type Communications

Abstract: Massive Machine Type Communication (mMTC) enables novel applications but its dense deployment and short packet properties lead to new challenges for physical layer design. This paper investigates hyper-dimensional modulation (HDM), a recently proposed novel non-orthogonal modulation, for short packet communications with superior interference tolerance in mMTC. We propose a new tree-based K-best decoding algorithm for HDM to improve the packet error rate performance in both additive white Gaussian noise (AWGN) and interference-limited scenarios. Simulation results show that the proposed algorithm can achieve 0.5 – 4 dB gain in AWGN and interference-limited channels compared to the Polar code with CRC (cyclic redundancy check)-aided list decoding.

Local List Recovery of High-Rate Tensor Codes and Applications

Abstract: We show that the tensor product of a high-rate globally list recoverable code is (approximately) locally list recoverable. List recovery has been a useful building block in the design of list decod...

Efficient List-Decoding with Constant Alphabet and List Sizes

Abstract: We present an explicit and efficient algebraic construction of capacity-achieving list decodable codes with both constant alphabet and constant list sizes. More specifically, for any $R \in (0,1)$ and $\epsilon>0$, we give an algebraic construction of an infinite family of error-correcting codes of rate $R$, over an alphabet of size $(1/\epsilon)^{O(1/\epsilon^2)}$, that can be list decoded from a $(1-R-\epsilon)$-fraction of errors with list size at most $\exp(\mathrm{poly}(1/\epsilon))$. Moreover, the codes can be encoded in time $\mathrm{poly}(1/\epsilon, n)$, the output list is contained in a linear subspace of dimension at most $\mathrm{poly}(1/\epsilon)$, and a basis for this subspace can be found in time $\mathrm{poly}(1/\epsilon, n)$. Thus, both encoding and list decoding can be performed in fully polynomial-time $\mathrm{poly}(1/\epsilon, n)$, except for pruning the subspace and outputting the final list which takes time $\exp(\mathrm{poly}(1/\epsilon))\cdot\mathrm{poly}(n)$. Our codes are quite natural and structured. Specifically, we use algebraic-geometric (AG) codes with evaluation points restricted to a subfield, and with the message space restricted to a (carefully chosen) linear subspace. Our main observation is that the output list of AG codes with subfield evaluation points is contained in an affine shift of the image of a block-triangular-Toeplitz (BTT) matrix, and that the list size can potentially be reduced to a constant by restricting the message space to a BTT evasive subspace, which is a large subspace that intersects the image of any BTT matrix in a constant number of points. We further show how to explicitly construct such BTT evasive subspaces, based on the explicit subspace designs of Guruswami and Kopparty (Combinatorica, 2016), and composition.

List Decodable Subspace Recovery

Abstract: Learning from data in the presence of outliers is a fundamental problem in statistics. In this work, we study robust statistics in the presence of overwhelming outliers for the fundamental problem of subspace recovery. Given a dataset where an $\alpha$ fraction (less than half) of the data is distributed uniformly in an unknown $k$ dimensional subspace in $d$ dimensions, and with no additional assumptions on the remaining data, the goal is to recover a succinct list of $O(\frac{1}{\alpha})$ subspaces one of which is nontrivially correlated with the planted subspace. We provide the first polynomial time algorithm for the 'list decodable subspace recovery' problem, and subsume it under a more general framework of list decoding over distributions that are "certifiably resilient" capturing state of the art results for list decodable mean estimation and regression.

Lattice (List) Decoding Near Minkowski's Inequality

Abstract: Minkowski proved that any $n$-dimensional lattice of unit determinant has a nonzero vector of Euclidean norm at most $\sqrt{n}$; in fact, there are $2^{\Omega(n)}$ such lattice vectors. Lattices whose minimum distances come close to Minkowski's bound provide excellent sphere packings and error-correcting codes in $\mathbb{R}^{n}$. The focus of this work is a certain family of efficiently constructible $n$-dimensional lattices due to Barnes and Sloane, whose minimum distances are within an $O(\sqrt{\log n})$ factor of Minkowski's bound. Our primary contribution is a polynomial-time algorithm that list decodes this family to distances approaching $1/\sqrt{2}$ of the minimum distance. The main technique is to decode Reed-Solomon codes under error measured in the Euclidean norm, using the Koetter-Vardy "soft decision" variant of the Guruswami-Sudan list-decoding algorithm.

A Deep Learning Assisted Node-Classified Redundant Decoding Algorithm for BCH Codes

Abstract: This paper proposes a node-classified redundant decoding (NC-RD) algorithm based on the received sequence’s channel reliability for high-density parity-check (HDPC) codes. Two preprocessing steps are proposed prior decoding. The variable nodes of the parity-check matrix are firstly classified by the $k$ -median algorithm based on the number of shortest cycles associated with each variable node before decoding. Then, by searching among the automorphism group of the HDPC codes, we generate a list of permutations for bit positions by computing and sorting the permutation reliability metrics. The redundant decoder conducts the message-passing decoding according to the sorted permutations, which limit the unreliable information propagation for each permutation. Besides proposing a list decoding algorithm on top of the NC-RD algorithm to augment the decoder’s performance, we show that the NC-RD algorithm can be transformed into a neural network system. More specifically, multiplicative tuneable weights are attached to the decoding messages to optimize the decoding performance. Simulation results of BCH codes over the AWGN channels show that the NC-RD algorithm provides a performance gain compared to the random redundant decoding algorithm. Additional decoding performance gain can be obtained by both the list decoding method and the neural network “learned” NC-RD algorithm.

Bounds for List-Decoding and List-Recovery of Random Linear Codes.

Abstract: A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input list size $\ell$ if for every sufficiently large subset of codewords (of size $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires list size $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input list size $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires list size $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with list size with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

Polarization-adjusted Convolutional (PAC) Codes: Sequential Decoding vs List Decoding

Abstract: In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Ar{\i}kan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes. In this scheme, a pair of polar mapper and demapper as pre- and postprocessing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER=10^-3 range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

Bounds for list-decoding and list-recovery of random linear codes

Abstract: A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input list size $\ell$ if for every sufficiently large subset of codewords (of size $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires list size $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input list size $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires list size $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with list size with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

Low Latency Decoder for Short Blocklength Polar Codes

Abstract: Polar codes have been gaining a lot of interest due to it being the first coding scheme to provably achieve the symmetric capacity of a binary memoryless channel with an explicit construction. However, the main drawback of polar codes is the low throughput of its successive cancellation (SC) decoding. Simplified SC decoding algorithms of polar codes can be used to reduce the latency of the polar decoder by faster processing of specific sub-codes in the polar code. By combining simplified SC with a list decoding technique, such as SC list (SCL) decoding, polar codes can cater to the two conflicting requirements of high reliability and low latency in ultra-reliable low-latency (URLLC) communication systems. Simplified SC algorithm recognises some special nodes in the SC decoding tree, corresponding to the specific subcodes in the polar code construction, and efficiently prunes the SC decoding tree, without traversing the sub-trees and computing log-likelihood ratios (LLRs) for each child node. However, this decoding process still suffers from the latency associated with the serial nature of SC decoding. We propose some new algorithms to process new types of node patterns that appear within multiple levels of pruned sub-trees and it enables to process certain nodes in parallel. In short blocklength polar codes, our proposed algorithms can achieve up to 13% latency reduction from fast-simplified SC [1] without any performance degradation. Furthermore, it can achieve up to 27% latency reduction if small error-correcting performance degradation is allowed.