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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. List-decodable codes are also considered in rank-metric, subspace metric, cover-metric, pair metric and insdel metric settings. 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 general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes of finite metric spaces. The general covering code upper bounds can apply to the case when the volumes of the balls depend on the centers, not only on the radius case. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the size of list-decodable codes.Our results give exponential improvements on the recent generalized Singleton upper bound in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are large. Even for the list size $L=1$ case our covering code upper bounds give highly non-trivial upper bounds on the sizes of codes with the given minimum distance.The generalized Singleton upper bound for average-radius list-decodable codes is given. The asymptotic forms of covering code bounds can partially recover the Blinovsky bound and the combinatorial bound of Guruswami-H{\aa}stad-Sudan-Zuckerman in Hamming metric setting. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes. We apply our general covering code upper bounds for list-decodable rank-metric codes, list-decodable subspace codes, list-decodable insertion codes and list-decodable deletion codes. Some new better results about non-list-decodability of rank-metric codes and subspace codes are obtained.

Continuous-time quantum walks have proven to be an extremely useful framework for the design of several quantum algorithms. Often, the running time of quantum algorithms in this framework is characterized by the quantum hitting time: the time required by the quantum walk to find a vertex of interest with a high probability. In this article, we provide improved upper bounds for the quantum hitting time that can be applied to several CTQW-based quantum algorithms. In particular, we apply our techniques to the glued-trees problem, improving their hitting time upper bound by a polynomial factor: from $O(n^5)$ to $O(n^2\log n)$. Furthermore, our methods also help to exponentially improve the dependence on precision of the continuous-time quantum walk based algorithm to find a marked node on any ergodic, reversible Markov chain by Chakraborty et al. [PRA 102, 022227 (2020)].

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. At first, we prove the fact mentioned in [A.E.Brouwer et al. Bounds on mixed binary/ternary codes. IEEE Trans. Inf. Theory 44 (1998) 140-161] that such codes are optimal, generalizing it to a bound for multifold packings of radius-$1$ balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$-fold packing with minimum distance $2$. At second, we show the existence of $4$-ary codes with parameters of shortened $1$-perfect codes that cannot be obtained by shortening a $1$-perfect code. Keywords: Hamming graph; multifold packings; multiple coverings; perfect codes.

PAC-Bayesian bounds are known to be tight and informative when studying the generalization ability of randomized classifiers. However, when applied to some family of deterministic models such as neural networks, they require a loose and costly derandomization step. As an alternative to this step, we introduce new PAC-Bayesian generalization bounds that have the originality to provide disintegrated bounds, i.e., they give guarantees over one single hypothesis instead of the usual averaged analysis. Our bounds are easily optimizable and can be used to design learning algorithms. We illustrate the interest of our result on neural networks and show a significant practical improvement over the state-of-the-art framework.

The 1993 Stern authentication protocol is a code-based zero-knowledge protocol with cheating probability equal to 2/3 based on the syndrome decoding problem which permits to obtain a proof of knowledge of a small weight vector. This protocol was improved a few years later by V\'eron, who proposed a variation of the scheme based on the general syndrome decoding problem which leads to better results in term of communication. A few years later, the AGS protocol introduced a variation of the V\'eron protocol based on quasi-cyclic matrices. The AGS protocol permits to obtain an asymptotic cheating probability of 1/2 and a strong improvement in term of communications. In the present paper we propose two new contributions. First, a Quasi-Cyclic Stern proof of knowledge construction which constitutes an adaptation of the AGS scheme in a syndrome decoding context. The main interest of this adaptation is that at the difference of the regular (non quasi-cyclic) case, the Quasi-Cyclic Stern protocol is better in terms of communication than its V\'eron counterpart (the AGS protocol, which can be seen as a Quasi-Cyclic V\'eron protocol). The difference comes from the fact that a seed related optimization is better for QC-Stern than for QC-V\'eron. Secondly, we also propose a general new optimization to handle random seeds in this type of protocol. Overall, the two new optimizations we propose permit to gain about 17.5% in the length of communication compared to the previous best approach for this type of protocols. Such optimizations are of great matter in the ongoing context where a new signature call for proposals has been announced by the NIST and for which such zero-knowledge approaches are a real alternative, as it was shown in the first signature call for proposals of the NIST.

This paper is {devoted} to efficient design of complete complementary codes (CCCs) which have found wide applications in coding, signal processing and wireless communication, thanks to their {ideal} auto- and cross-correlation sum properties. A major motivation of this research is that the existing state-of-the-art can generate CCCs with certain lengths only and therefore may not {be able to meet} the diverse requirements in practice. We introduce a new tool called multivariable functions {with which we} propose a direct construction of CCCs with any \textit{arbitrary} lengths in the form of $\prod_{i=1}^k p_i^{m_i}$, where $k$ is a positive integer, $p_1,p_2,\hdots,p_k$ are prime numbers, and $m_1,m_2,\hdots,m_k$ are positive integers. The proposed set size is given by $\prod_{i=1}^k p_i^{n_i}$ ($n_1,n_2,\hdots,n_k$ positive integers), which covers all possible positive integers greater than 1. For $k=1$ and $p_1=2$, our proposed {construction reduces} to the exact Golay-Davis-Jedwab (GDJ) sequence generator as a special case. For $k>1$ and $p_1=p_2=\cdots=p_k=2$, it gives rise to the conventional CCCs with power-of-two lengths obtained from generalized Boolean functions. {Moreover, we introduce a linear code in connection with the proposed sets of CCCs.}

We argue that proven exponential upper bounds on runtimes, an established area in classic algorithms, are interesting also in heuristic search and we prove several such results. We show that any of the algorithms randomized local search, Metropolis algorithm, simulated annealing, and (1+1) evolutionary algorithm can optimize any pseudo-Boolean weakly monotonic function under a large set of noise assumptions in a runtime that is at most exponential in the problem dimension~$n$. This drastically extends a previous such result, limited to the (1+1) EA, the LeadingOnes function, and one-bit or bit-wise prior noise with noise probability at most $1/2$, and at the same time simplifies its proof. With the same general argument, among others, we also derive a sub-exponential upper bound for the runtime of the $(1,\lambda)$ evolutionary algorithm on the OneMax problem when the offspring population size $\lambda$ is logarithmic, but below the efficiency threshold. To show that our approach can also deal with non-trivial parent population sizes, we prove an exponential upper bound for the runtime of the mutation-based version of the simple genetic algorithm on the OneMax benchmark, matching a known exponential lower bound.

There have been significant research activities in recent years to automate the design of channel encoders and decoders via deep learning. Due the dimensionality challenge in channel coding, it is prohibitively complex to design and train relatively large neural channel codes via deep learning techniques. Consequently, most of the results in the literature are limited to relatively short codes having less than 100 information bits. In this paper, we construct ProductAEs, a computationally efficient family of deep-learning driven (encoder, decoder) pairs, that aim at enabling the training of relatively large channel codes (both encoders and decoders) with a manageable training complexity. We build upon the ideas from classical product codes, and propose constructing large neural codes using smaller code components. More specifically, instead of directly training the encoder and decoder for a large neural code of dimension $k$ and blocklength $n$, we provide a framework that requires training neural encoders and decoders for the code parameters $(k_1,n_1)$ and $(k_2,n_2)$ such that $k_1 k_2=k$ and $n_1 n_2=n$. Our training results show significant gains, over all ranges of signal-to-noise ratio (SNR), for a code of parameters $(100,225)$ and a moderate-length code of parameters $(196,441)$, over polar codes under successive cancellation (SC) decoder. Moreover, our results demonstrate meaningful gains over Turbo Autoencoder (TurboAE) and state-of-the-art classical codes. This is the first work to design product autoencoders and a pioneering work on training large channel codes.

We study financial networks with debt contracts and credit default swaps between specific pairs of banks. Given such a financial system, we want to decide which of the banks are in default, and how much of their liabilities can these defaulting banks pay. There can easily be multiple different solutions to this problem, leading to a situation of default ambiguity, and a range of possible solutions to implement for a financial authority. In this paper, we study the properties of the solution space of such financial systems, and analyze a wide range of reasonable objective functions for selecting from the set of solutions. Examples of such objective functions include minimizing the number of defaulting banks, minimizing the amount of unpaid debt, maximizing the number of satisfied banks, and many others. We show that for all of these objectives, it is NP-hard to approximate the optimal solution to an $n^{1-\epsilon}$ factor for any $\epsilon>0$, with $n$ denoting the number of banks. Furthermore, we show that this situation is rather difficult to avoid from a financial regulator's perspective: the same hardness results also hold if we apply strong restrictions on the weights of the debts, the structure of the network, or the amount of funds that banks must possess. However, if we restrict both the network structure and the amount of funds simultaneously, then the solution becomes unique, and it can be found efficiently.

A rep-tile is a polygon that can be dissected into smaller copies (of the same size) of the original polygon. A polyomino is a polygon that is formed by joining one or more unit squares edge to edge. These two notions were first introduced and investigated by Solomon W. Golomb in the 1950s and popularized by Martin Gardner in the 1960s. Since then, dozens of studies have been made in communities of recreational mathematics and puzzles. In this study, we first focus on the specific rep-tiles that have been investigated in these communities. Since the notion of rep-tiles is so simple that can be formulated mathematically in a natural way, we can apply a representative puzzle solver, a MIP solver, and SAT-based solvers for solving the rep-tile problem in common. In comparing their performance, we can conclude that the puzzle solver is the weakest while the SAT-based solvers are the strongest in the context of simple puzzle solving. We then turn to analyses of the specific rep-tiles. Using some properties of the rep-tile patterns found by a solver, we can complete analyses of specific rep-tiles up to certain sizes. That is, up to certain sizes, we can determine the existence of solutions, clarify the number of the solutions, or we can enumerate all the solutions for each size. In the last case, we find new series of solutions for the rep-tiles which have never been found in the communities.