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We consider the problem of coded distributed computing using polar codes. The average execution time of a coded computing system is related to the error probability for transmission over the binary erasure channel in recent work by Soleymani, Jamali and Mahdavifar, where the performance of binary linear codes is investigated. In this paper, we focus on polar codes and unveil a connection between the average execution time and the scaling exponent $\mu$ of the family of codes. The scaling exponent has emerged as a central object in the finite-length characterization of polar codes, and it captures the speed of convergence to capacity. In particular, we show that (i) the gap between the normalized average execution time of polar codes and that of optimal MDS codes is $O(n^{-1/\mu})$, and (ii) this upper bound can be improved to roughly $O(n^{-1/2})$ by considering polar codes with large kernels. We conjecture that these bounds could be improved to $O(n^{-2/\mu})$ and $O(n^{-1})$, respectively, and provide a heuristic argument as well as numerical evidence supporting this view.

We prove two theorems related to the Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make assertions about the behavior of a single random sequence, but only on the distributional behavior of a sequence of random variables. Semantically, we usually interpret CLTs as assertions about the collective behavior of infinitely many sequences. Yet, our intuition is that if a sequence of bits is "truly random", then it should provide a "source of randomness" for which CLT-type results should hold. We tackle this difficulty by using a sampling scheme that generates an infinite number of samples from a single binary sequence. We show that when we apply this scheme to a Martin-L\"of random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution. We also prove the well known almost sure central limit theorem (ASCLT), which provides an alternative, albeit less intuitive, answer to this question. Both results are also generalized for Schnorr random sequences.

A locally testable code (LTC) is an error correcting code with a property tester. The tester tests if a word is codeword by reading constant random bits and rejects the word with probability proportional to the distance from the word to the closest codeword. An important open question until recently is whether there exist $c^3$-LTCs which are LTCs with constant rate, constant relative distance and constant locality. In this work, we construct a new LTC family using 1-sided lossless expanders and balanced products.

In this work, a sequential decoder for convolutional codes over channels that are vulnerable to insertion, deletion, and substitution errors, is described and analyzed. The decoder expands the code trellis by introducing a new channel state variable, called drift state, as proposed by Davey-MacKay. A suitable decoding metric on that trellis for sequential decoding is derived, in a manner that generalizes the original Fano metric. Under low-noise environments, this approach reduces the decoding complexity by a couple orders of magnitude in comparison to Viterbi's algorithm, albeit at relatively higher frame error rates. An analytical method to determine the computational cutoff rate is also suggested. This analysis is supported with numerical evaluations of frame error rates and computational complexity, which are compared with respect to optimal Viterbi decoding.

Sum-of-norms clustering is a popular convexification of $K$-means clustering. We show that, if the dataset is made of a large number of independent random variables distributed according to the uniform measure on the union of two disjoint balls of unit radius, and if the balls are sufficiently close to one another, then sum-of-norms clustering will typically fail to recover the decomposition of the dataset into two clusters. As the dimension tends to infinity, this happens even when the distance between the centers of the two balls is taken to be as large as $2\sqrt{2}$. In order to show this, we introduce and analyze a continuous version of sum-of-norms clustering, where the dataset is replaced by a general measure. In particular, we state and prove a local-global characterization of the clustering that seems to be new even in the case of discrete datapoints.

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.

This paper is concerned with list decoding of $2$-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. The decoding radius exceeds the binary Johnson radius and the newly derived upper bound on the returned list size scales polynomially in the code parameters. Further, we provide simulation results on the probability of successful decoding by the proposed algorithm.

This paper studies the adversarial torn-paper channel. This problem is motivated by applications in DNA data storage where the DNA strands that carry the information may break into smaller pieces that are received out of order. Our model extends the previously researched probabilistic setting to the worst-case. We develop code constructions for any parameters of the channel for which non-vanishing asymptotic rate is possible and show our constructions achieve optimal asymptotic rate while allowing for efficient encoding and decoding. Finally, we extend our results to related settings included multi-strand storage, presence of substitution errors, or incomplete coverage.

In this paper, we present an explicit construction of list-decodable codes for single-deletion and single-substitution with list size two and redundancy 3log n+4, where n is the block length of the code. Our construction has lower redundancy than the best known explicit construction by Gabrys et al. (arXiv 2021), whose redundancy is 4log n+O(1).

As a natural language generation task, it is challenging to generate informative and coherent review text. In order to enhance the informativeness of the generated text, existing solutions typically learn to copy entities or triples from knowledge graphs (KGs). However, they lack overall consideration to select and arrange the incorporated knowledge, which tends to cause text incoherence. To address the above issue, we focus on improving entity-centric coherence of the generated reviews by leveraging the semantic structure of KGs. In this paper, we propose a novel Coherence Enhanced Text Planning model (CETP) based on knowledge graphs (KGs) to improve both global and local coherence for review generation. The proposed model learns a two-level text plan for generating a document: (1) the document plan is modeled as a sequence of sentence plans in order, and (2) the sentence plan is modeled as an entity-based subgraph from KG. Local coherence can be naturally enforced by KG subgraphs through intra-sentence correlations between entities. For global coherence, we design a hierarchical self-attentive architecture with both subgraph- and node-level attention to enhance the correlations between subgraphs. To our knowledge, we are the first to utilize a KG-based text planning model to enhance text coherence for review generation. Extensive experiments on three datasets confirm the effectiveness of our model on improving the content coherence of generated texts.