We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.
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Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes.
In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter $r$ in such codes by using only non-rational places of degrees at most $r$. This is, up to the author's knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over $\mathbb F_3$. We then investigate similarities with certain concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia-Stichtenoth tower of function fields, for both our construction and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our new approach.
Traditional dataset retrieval systems index on metadata information rather than on the data values. Thus relying primarily on manual annotations and high-quality metadata, processes known to be labour-intensive and challenging to automate. We propose a method to support metadata enrichment with topic annotations of column headers using three Large Language Models (LLMs): ChatGPT-3.5, GoogleBard and GoogleGemini. We investigate the LLMs ability to classify column headers based on domain-specific topics from a controlled vocabulary. We evaluate our approach by assessing the internal consistency of the LLMs, the inter-machine alignment, and the human-machine agreement for the topic classification task. Additionally, we investigate the impact of contextual information (i.e. dataset description) on the classification outcomes. Our results suggest that ChatGPT and GoogleGemini outperform GoogleBard for internal consistency as well as LLM-human-alignment. Interestingly, we found that context had no impact on the LLMs performances. This work proposes a novel approach that leverages LLMs for text classification using a controlled topic vocabulary, which has the potential to facilitate automated metadata enrichment, thereby enhancing dataset retrieval and the Findability, Accessibility, Interoperability and Reusability (FAIR) of research data on the Web.
By abstracting over well-known properties of De Bruijn's representation with nameless dummies, we design a new theory of syntax with variable binding and capture-avoiding substitution. We propose it as a simpler alternative to Fiore, Plotkin, and Turi's approach, with which we establish a strong formal link. We also show that our theory easily incorporates simple types and equations between terms.
The goal of Universal Cross-Domain Retrieval (UCDR) is to achieve robust performance in generalized test scenarios, wherein data may belong to strictly unknown domains and categories during training. Recently, pre-trained models with prompt tuning have shown strong generalization capabilities and attained noteworthy achievements in various downstream tasks, such as few-shot learning and video-text retrieval. However, applying them directly to UCDR may not sufficiently to handle both domain shift (i.e., adapting to unfamiliar domains) and semantic shift (i.e., transferring to unknown categories). To this end, we propose \textbf{Pro}mpting-to-\textbf{S}imulate (ProS), the first method to apply prompt tuning for UCDR. ProS employs a two-step process to simulate Content-aware Dynamic Prompts (CaDP) which can impact models to produce generalized features for UCDR. Concretely, in Prompt Units Learning stage, we introduce two Prompt Units to individually capture domain and semantic knowledge in a mask-and-align way. Then, in Context-aware Simulator Learning stage, we train a Content-aware Prompt Simulator under a simulated test scenarios to produce the corresponding CaDP. Extensive experiments conducted on three benchmark datasets show that our method achieves new state-of-the-art performance without bringing excessive parameters. Our method is publicly available at //github.com/fangkaipeng/ProS.
In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $\kappa\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $\lambda$ and on $\kappa$. Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $\lambda\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{\kappa+1}-1$ solutions for sufficiently negative $\lambda$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $\lambda$ and the profiles of positive solutions. Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.
Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let $C(n,q)$ be the maximum size of $q$-ary fixed-length non-overlapping codes of length $n$. The upper bound on $C(n,q)$ has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes of length at most $n$ remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a $q$-ary variable-length non-overlapping code is upper bounded by $C(n,q)$. Furthermore, we prove that the average length of the codewords in a $q$-ary variable-length non-overlapping codes is lower bounded by $\lceil \log_q \tilde{C} \rceil$, and is asymptotically no shorter than $n-2$ as $q$ approaches $\infty$, where $\tilde{C}$ denotes the cardinality of $q$-ary variable-length non-overlapping codes of length up to $n$.
We study Langevin-type algorithms for sampling from Gibbs distributions such that the potentials are dissipative and their weak gradients have finite moduli of continuity not necessarily convergent to zero. Our main result is a non-asymptotic upper bound of the 2-Wasserstein distance between a Gibbs distribution and the law of general Langevin-type algorithms based on the Liptser--Shiryaev theory and Poincar\'{e} inequalities. We apply this bound to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy if the potentials are dissipative and their gradients are uniformly continuous. We also propose Langevin-type algorithms with spherical smoothing for distributions whose potentials are not convex or continuously differentiable.
In this work, we present new constructions for topological subsystem codes using semi-regular Euclidean and hyperbolic tessellations. They give us new families of codes, and we also provide a new family of codes obtained through an already existing construction, due to Sarvepalli and Brown. We also prove new results that allow us to obtain the parameters of these new codes.
A simple, recently observed generalization of the classical Singleton bound to list-decoding asserts that rate $R$ codes are not list-decodable using list-size $L$ beyond an error fraction $\frac{L}{L+1} (1-R)$ (the Singleton bound being the case of $L=1$, i.e., unique decoding). We prove that in order to approach this bound for any fixed $L >1$, one needs exponential alphabets. Specifically, for every $L>1$ and $R\in(0,1)$, if a rate $R$ code can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$, then its alphabet must have size at least $\exp(\Omega_{L,R}(1/\varepsilon))$. This is in sharp contrast to the situation for unique decoding where certain families of rate $R$ algebraic-geometry (AG) codes over an alphabet of size $O(1/\varepsilon^2)$ are unique-decodable up to error fraction $(1-R-\varepsilon)/2$. Our bounds hold even for subconstant $\varepsilon\ge 1/n$, implying that any code exactly achieving the $L$-th generalized Singleton bound requires alphabet size $2^{\Omega_{L,R}(n)}$. Previously this was only known only for $L=2$ under the additional assumptions that the code is both linear and MDS. Our lower bound is tight up to constant factors in the exponent -- with high probability random codes (or, as shown recently, even random linear codes) over $\exp(O_L(1/\varepsilon))$-sized alphabets, can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$.
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