The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. There are general result about the Johnson radius and the list-decoding capacity theorem for random codes. However few results about general constraints on rates, list-decodable radius and list sizes for list-decodable codes have been obtained. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new simple but strong upper bounds for list-decodable codes based on various covering codes. Then any good upper bound on the covering radius imply a good upper bound on the size of list-decodable codes. Hence the list-decodablity of codes is a strong constraint from the view of covering codes. Our covering code upper bounds for $(d,1)$ list decodable codes give highly non-trivial upper bounds on the sizes of codes with the given minimum Hamming distances. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020, when the code lengths are very large. The asymptotic forms of covering code bounds can partially recover the list-decoding capacity theorem, the Blinovsky bound and the combinatorial bound of Guruswami-H{\aa}stad-Sudan-Zuckerman. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes.
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The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive. This in turn guarantees the existence of a quantum error correction code for noise modeled by that channel. Discretizing the single-qubit errors leads to the important family of Pauli quantum channels; curiously, multipartite entangled states can increase the threshold of these channels beyond the so-called hashing bound, an effect termed superadditivity of coherent information. In this work, we divide the simplex of Pauli channels into one-parameter families and compute numerical lower bounds on their error thresholds. We find substantial increases of error thresholds relative to the hashing bound for large regions in the Pauli simplex corresponding to biased noise, which is a realistic noise model in promising quantum computing architectures. The error thresholds are computed on the family of graph states, a special type of stabilizer state. In order to determine the coherent information of a graph state, we devise an algorithm that exploits the symmetries of the underlying graph, resulting in a substantial computational speed-up. This algorithm uses tools from computational group theory and allows us to consider symmetric graph states on a large number of vertices. Our algorithm works particularly well for repetition codes and concatenated repetition codes (or cat codes), for which our results provide the first comprehensive study of superadditivity for arbitrary Pauli channels. In addition, we identify a novel family of quantum codes based on tree graphs. The error thresholds of these tree graph states outperform repetition and cat codes in large regions of the Pauli simplex, and hence form a new code family with desirable error correction properties.
We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than the known generic decoders for the Hamming metric and smaller than the known rank-metric decoders. Furthermore, we give a formal hardness reduction, providing evidence that generic sum-rank decoding is computationally hard. As a by-product of the above, we solve some fundamental coding problems in the sum-rank metric: we give an algorithm to compute the exact size of a sphere of a given sum-rank radius, and also give an upper bound as a closed formula; and we study erasure decoding with respect to two different notions of support.
In this paper, we design the joint decoding (JD) of non-orthogonal multiple access (NOMA) systems employing short block length codes. We first proposed a low-complexity soft-output ordered-statistics decoding (LC-SOSD) based on a decoding stopping condition, derived from approximations of the a-posterior probabilities of codeword estimates. Simulation results show that LC-SOSD has the similar mutual information transform property to the original SOSD with a significantly reduced complexity. Then, based on the analysis, an efficient JD receiver which combines the parallel interference cancellation (PIC) and the proposed LC-SOSD is developed for NOMA systems. Two novel techniques, namely decoding switch (DS) and decoding combiner (DC), are introduced to accelerate the convergence speed. Simulation results show that the proposed receiver can achieve a lower bit-error rate (BER) compared to the successive interference cancellation (SIC) decoding over the additive-white-Gaussian-noise (AWGN) and fading channel, with a lower complexity in terms of the number of decoding iterations.
Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent homology modules. We address the inverse problem: given a target barcode $D$, computing the fiber $\mathrm{PH}^{-1}(D)$. For this, we use the fact that $\mathrm{PH}^{-1}(D)$ decomposes as complex of polyhedra when $K$ is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth first search algorithm that recovers the polyhedra forming the fiber $\mathrm{PH}^{-1}(D)$. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.
This paper proposes new polar code design principles for the low-latency automorphism ensemble (AE) decoding. Our proposal permits to design a polar code with the desired automorphism group (if possible) while assuring the decreasing monomial property. Moreover, we prove that some automorphisms are redundant under AE decoding, and we propose a new automorphisms classification based on equivalence classes. Finally, we propose an automorphism selection heuristic based on drawing only one element of each class; we show that this method enhances the block error rate (BLER) performance of short polar codes even with a limited number of automorphisms.
Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem in computational chemistry. Due to the exponentially large search space, the problem has been referred in several materials-science papers as "NP-Hard and very challenging" without any formal proof though. This paper fills a gap in the literature providing the first set of formally proven NP-Hardness results for a variant of csp with various realistic constraints. In particular, we focus on the problem of removal: the goal is to find a substructure with minimal potential energy, by removing a subset of the ions from a given initial structure. Our main contributions are NP-Hardness results for the csp removal problem, new embeddings of combinatorial graph problems into geometrical settings, and a more systematic exploration of the energy function to reveal the complexity of csp. In a wider context, our results contribute to the analysis of computational problems for weighted graphs embedded into the three-dimensional Euclidean space.
Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square q, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of q-ary MDS Euclidean self-dual codes. In particular, for large square q, our constructions take up a proportion of generally more than 34% in all the possible lengths of q-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.
Recently, the information-theoretical framework has been proven to be able to obtain non-vacuous generalization bounds for large models trained by Stochastic Gradient Langevin Dynamics (SGLD) with isotropic noise. In this paper, we optimize the information-theoretical generalization bound by manipulating the noise structure in SGLD. We prove that with constraint to guarantee low empirical risk, the optimal noise covariance is the square root of the expected gradient covariance if both the prior and the posterior are jointly optimized. This validates that the optimal noise is quite close to the empirical gradient covariance. Technically, we develop a new information-theoretical bound that enables such an optimization analysis. We then apply matrix analysis to derive the form of optimal noise covariance. Presented constraint and results are validated by the empirical observations.
Generative models have been successfully used for generating realistic signals. Because the likelihood function is typically intractable in most of these models, the common practice is to use "implicit" models that avoid likelihood calculation. However, it is hard to obtain theoretical guarantees for such models. In particular, it is not understood when they can globally optimize their non-convex objectives. Here we provide such an analysis for the case of Maximum Mean Discrepancy (MMD) learning of generative models. We prove several optimality results, including for a Gaussian distribution with low rank covariance (where likelihood is inapplicable) and a mixture of Gaussians. Our analysis shows that that the MMD optimization landscape is benign in these cases, and therefore gradient based methods will globally minimize the MMD objective.
We address the question of characterizing and finding optimal representations for supervised learning. Traditionally, this question has been tackled using the Information Bottleneck, which compresses the inputs while retaining information about the targets, in a decoder-agnostic fashion. In machine learning, however, our goal is not compression but rather generalization, which is intimately linked to the predictive family or decoder of interest (e.g. linear classifier). We propose the Decodable Information Bottleneck (DIB) that considers information retention and compression from the perspective of the desired predictive family. As a result, DIB gives rise to representations that are optimal in terms of expected test performance and can be estimated with guarantees. Empirically, we show that the framework can be used to enforce a small generalization gap on downstream classifiers and to predict the generalization ability of neural networks.
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