arXiv subjects

Van Khu Vu

Publications and source records attributed to Van Khu Vu.

At least 19 recordsLinked to original sources

arXiv2026

Near-Optimal Covering Sequences

An $(n,R)$-covering sequence over a finite alphabet $Σ_q = \{0,1,\dots, q-1\}$ is a cyclic sequence whose consecutive length-$n$ windows form a covering code of radius $R$. Equivalently, every word in $Σ_q^n$ is within Hamming distance $R$ of at least one window. We give a deterministic and explicit construction of such sequences whose length, for every fixed alphabet size $q$, every fixed radius $R$, and every sufficiently large $n$, attains the sphere-covering lower bound up to a constant factor depending only on $q$ and $R$. Thus, in the fixed-radius regime, the construction removes the logarithmic factor in the general probabilistic upper bounds of [Chung and Cooper, \emph{Random Structures \& Algorithms}, 2004] and [Vu, \emph{Advances in Applied Mathematics}, 2005]. It also complements the earlier explicit constructions of [Chee, Etzion, Ta, and Vu, \emph{Designs, Codes and Cryptography}, 2025], which include constant factor bounds for the special binary radius-one families $n=2^a-1$ and $n=2^a$, where $a\ge1$.

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arXiv2026

Block Tensor Rank of Sum-Rank Metric Codes

Sum-rank codes provide a generalized framework for Hamming and rank-metric codes, with codewords represented as tuples of matrices and weight given by the sum of the block ranks. In this paper, we introduce and study a block-tensor-rank invariant for sum-rank metric codes. To each code, we associate its \emph{block tensor rank}: the smallest number of block-simple tensors, namely rank-one matrices supported inside single blocks, whose linear span contains the code. In general, determining the block tensor rank of a sum-rank code is challenging. Our main structural result shows that the block tensor rank decomposes additively across the blocks of the code, thereby reducing its computation to a tensor-rank problem on each block projection. Consequently, we derive two complementary lower bounds on the block tensor rank, referred to as the \emph{projection-wise bound} and the \emph{coordinate-code bound}. Moreover, by combining the coordinate-code bound with the classical Singleton and Griesmer bounds for codes in the Hamming metric, we obtain explicit lower bounds, called the \emph{Singleton coordinate-code bound} and the \emph{Griesmer coordinate-code bound}, respectively. We further construct families of sum-rank codes whose block tensor ranks attain the Singleton or Griesmer coordinate-code bounds. These constructions are based on Hamming-metric codes achieving the corresponding classical bounds. Finally, we show that, in certain cases, the block tensor ranks of two known families of sum-rank codes in the literature do not attain the Singleton coordinate-code bound.

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arXiv2026

Beyond Static Priors: Dynamic Neural Guidance for Large-Scale Ant Colony Optimization

Neural-guided Ant Colony Optimization (ACO) suffers from a fundamental training-inference misalignment: policies are typically trained to generate static priors (e.g., heatmaps), yet deployed to guide iterative, long-horizon search processes. In this paper, we present DyNACO, a novel framework that achieves dynamic neural guidance by periodically observing the pheromone distribution and the incumbent solution. To make DyNACO tractable at scale, we pair the policy with a perturbation-based ACO backend and a scope-restricted refinement mechanism that jointly ensure efficacy and stable credit assignment. On TSP, DyNACO scales to 100,000-node instances and outperforms neural baselines while often reducing total runtime compared to the unguided solver. We extend DyNACO to CVRP via a capacity-aware backend, consistently improving the unguided baseline with less than 1% neural overhead. We further provide in-depth analysis validating the model's generalization capabilities and elucidating why dynamic guidance outperforms static priors. Our work underscores the necessity of aligning neural training with iterative search dynamics in learning-guided optimization. The code is available at https://github.com/shoraaa/DyNACO.

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arXiv2026

List Reconstruction Problem with List Size Two

The problem of computing the cardinality of the intersection of multiple balls in the Hamming space has attracted a lot of attention recently due to their applications in the list reconstruction problem and information retrieval in Associative Memories. In previous work, most of the results are for the cases where the radii of each ball, $r$ and the distance between the centers of these balls, $k$ are fixed when the length $n$ of each codeword tend to infinity. In this work, we focus on the case where $r = αn$ and $k=βn$ for some constants $α$ and $β$ and compute the maximum asymptotic rate of the cardinality of the intersection of three balls. We provide the maximum asymptotic rate as a function of two parameters $α$ and $β$. We also provide numerical results and compare these results with the intersection of two balls.

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arXiv2026

A Mixture of Experts Vision Transformer for High-Fidelity Surface Code Decoding

Quantum error correction is a key ingredient for large scale quantum computation, protecting logical information from physical noise by encoding it into many physical qubits. Topological stabilizer codes are particularly appealing due to their geometric locality and practical relevance. In these codes, stabilizer measurements yield a syndrome that must be decoded into a recovery operation, making decoding a central bottleneck for scalable real time operation. Existing decoders are commonly classified into two categories. Classical algorithmic decoders provide strong and well established baselines, but may incur substantial computational overhead at large code distances or under stringent latency constraints. Machine learning based decoders offer fast GPU inference and flexible function approximation, yet many approaches do not explicitly exploit the lattice geometry and local structure of topological codes, which can limit performance. In this work, we propose QuantumSMoE, a quantum vision transformer based decoder that incorporates code structure through plus shaped embeddings and adaptive masking to capture local interactions and lattice connectivity, and improves scalability via a mixture of experts layer with a novel auxiliary loss. Experiments on the toric code demonstrate that QuantumSMoE outperforms state-of-the-art machine learning decoders as well as widely used classical baselines.

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arXiv2026

Sequence Reconstruction for Sticky Insertion/Deletion Channels

The sequence reconstruction problem for insertion/deletion channels has attracted significant attention owing to their applications recently in some emerging data storage systems, such as racetrack memories, DNA-based data storage. Our goal is to investigate the reconstruction problem for sticky-insdel channels where both sticky-insertions and sticky-deletions occur. If there are only sticky-insertion errors, the reconstruction problem for sticky-insertion channel is a special case of the reconstruction problem for tandem-duplication channel which has been well-studied. In this work, we consider the $(t, s)$-sticky-insdel channel where there are at most $t$ sticky-insertion errors and $s$ sticky-deletion errors when we transmit a message through the channel. For the reconstruction problem, we are interested in the minimum number of distinct outputs from these channels that are needed to uniquely recover the transmitted vector. We first provide a recursive formula to determine the minimum number of distinct outputs required. Next, we provide an efficient algorithm to reconstruct the transmitted vector from erroneous sequences.

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arXiv2026

Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $ε$ in $\mathrm{poly}(1/ε, \text{input dimension})$ time.

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arXiv2025

NeuFACO: Neural Focused Ant Colony Optimization for Traveling Salesman Problem

This study presents Neural Focused Ant Colony Optimization (NeuFACO), a non-autoregressive framework for the Traveling Salesman Problem (TSP) that combines advanced reinforcement learning with enhanced Ant Colony Optimization (ACO). NeuFACO employs Proximal Policy Optimization (PPO) with entropy regularization to train a graph neural network for instance-specific heuristic guidance, which is integrated into an optimized ACO framework featuring candidate lists, restricted tour refinement, and scalable local search. By leveraging amortized inference alongside ACO stochastic exploration, NeuFACO efficiently produces high-quality solutions across diverse TSP instances.

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arXiv2025

A New Construction of Non-Binary Deletion Correcting Codes and their Decoding

Non-binary codes correcting multiple deletions have recently attracted a lot of attention. In this work, we focus on multiplicity-free codes, a family of non-binary codes where all symbols are distinct. Our main contribution is a new explicit construction of such codes, based on set and permutation codes. We show that our multiplicity-free codes can correct multiple deletions and provide a decoding algorithm. We also show that, for a certain regime of parameters, our constructed codes have size larger than all the previously known non-binary codes correcting multiple deletions.

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arXiv2025

Constructions of Covering Sequences and Arrays

An $(n,R)$-covering sequence is a cyclic sequence whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. Using several construction methods improvements of the upper bounds on the length of such sequences for $n \leq 20$ and $1 \leq R \leq 3$, are obtained. The definition is generalized in two directions. An $(n,m,R)$-covering sequence code is a set of cyclic sequences of length $m$ whose consecutive $n$-tuples form a code of length~$n$ and covering radius $R$. The definition is also generalized to arrays in which the $m \times n$ sub-matrices form a covering code with covering radius $R$. We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.

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arXiv2024

Permutation and Multi-permutation Codes Correcting Multiple Deletions

Permutation codes in the Ulam metric, which can correct multiple deletions, have been investigated extensively recently. In this work, we are interested in the maximum size of permutation codes in the Ulam metric and aim to design permutation codes that can correct multiple deletions with efficient decoding algorithms. We first present an improvement on the Gilbert--Varshamov bound of the maximum size of these permutation codes by analyzing the independence number of the auxiliary graph. The idea is widely used in various cases and our contribution in this section is enumerating the number of triangles in the auxiliary graph and showing that it is small enough. Next, we design permutation codes correcting multiple deletions with a decoding algorithm. In particular, the constructed permutation codes can correct $t$ deletions with at most $(3t-1) \log n+o(\log n)$ bits of redundancy where $n$ is the length of the code. Our construction is based on a new mapping which yields a new connection between permutation codes in the Hamming metric and permutation codes in various metrics. Furthermore, we construct permutation codes that correct multiple bursts of deletions using this new mapping. Finally, we extend the new mapping for multi-permutations and construct the best-known multi-permutation codes in Ulam metric.

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arXiv2024

Maximum Length RLL Sequences in de Bruijn Graph

Free-space quantum key distribution requires to synchronize the transmitted and received signals. A timing and synchronization system for this purpose based on a de Bruijn sequence has been proposed and studied recently for a channel associated with quantum communication that requires reliable synchronization. To avoid a long period of no-pulse in such a system on-off pulses are used to simulate a \emph{zero} and on-on pulses are used to simulate a \emph{one}. However, these sequences have high redundancy and low rate. To reduce the redundancy and increase the rate, run-length limited sequences in the de Bruijn graph are proposed for the same purpose. The maximum length of such sequences in the de Bruijn graph is studied and an efficient algorithm to construct a large set of these sequences is presented. Based on known algorithms and enumeration methods, maximum length sequence for which the position of each window can be computed efficiently is presented and an enumeration on the number of such sequences is given.

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arXiv2024

On de Bruijn Covering Sequences and Arrays

An $(m,n,R)$-de Bruijn covering array (dBCA) is a doubly periodic $M \times N$ array over an alphabet of size $q$ such that the set of all its $m \times n$ windows form a covering code with radius $R$. An upper bound of the smallest array area of an $(m,n,R)$-dBCA is provided using a probabilistic technique which is similar to the one that was used for an upper bound on the length of a de Bruijn covering sequence. A folding technique to construct a dBCA from a de Bruijn covering sequence or de Bruijn covering sequences code is presented. Several new constructions that yield shorter de Bruijn covering sequences and $(m,n,R)$-dBCAs with smaller areas are also provided. These constructions are mainly based on sequences derived from cyclic codes, self-dual sequences, primitive polynomials, an interleaving technique, folding, and mutual shifts of sequences with the same covering radius. Finally, constructions of de Bruijn covering sequences codes are also discussed.

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arXiv2024

Efficient designs for threshold group testing without gap

Given $d$ defective items in a population of $n$ items with $d \ll n$, in threshold group testing without gap, the outcome of a test on a subset of items is positive if the subset has at least $u$ defective items and negative otherwise, where $1 \leq u \leq d$. The basic goal of threshold group testing is to quickly identify the defective items via a small number of tests. In non-adaptive design, all tests are designed independently and can be performed in parallel. The decoding time in the non-adaptive state-of-the-art work is a polynomial of $(d/u)^u (d/(d-u))^{d - u}, d$, and $\log{n}$. In this work, we present a novel design that significantly reduces the number of tests and the decoding time to polynomials of $\min\{u^u, (d - u)^{d - u}\}, d$, and $\log{n}$. In particular, when $u$ is a constant, the number of tests and the decoding time are $O(d^3 (\log^2{n}) \log{(n/d)} )$ and $O\big(d^3 (\log^2{n}) \log{(n/d)} + d^2 (\log{n}) \log^3{(n/d)} \big)$, respectively. For a special case when $u = 2$, with non-adaptive design, the number of tests and the decoding time are $O(d^3 (\log{n}) \log{(n/d)} )$ and $O(d^2 (\log{n} + \log^2{(n/d)}) )$, respectively. Moreover, with 2-stage design, the number of tests and the decoding time are $O(d^2 \log^2{(n/d)} )$.

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arXiv2023

Codes for Correcting Asymmetric Adjacent Transpositions and Deletions

Codes in the Damerau--Levenshtein metric have been extensively studied recently owing to their applications in DNA-based data storage. In particular, Gabrys, Yaakobi, and Milenkovic (2017) designed a length-$n$ code correcting a single deletion and $s$ adjacent transpositions with at most $(1+2s)\log n$ bits of redundancy. In this work, we consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions may occur. We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting a single deletion, $s^+$ right-shift, and $s^-$ left-shift errors with at most $(1+s)\log (n+s+1)+1$ bits of redundancy where $s=s^{+}+s^{-}$. In addition, we investigate codes correcting $t$ $0$-deletions, $s^+$ right-shift, and $s^-$ left-shift errors with both uniquely-decoding and list-decoding algorithms. Our main contribution here is the construction of a list-decodable code with list size $O(n^{\min\{s+1,t\}})$ and with at most $(\max \{t,s+1\}) \log n+O(1)$ bits of redundancy, where $s=s^{+}+s^{-}$. Finally, we construct both non-systematic and systematic codes for correcting blocks of $0$-deletions with $\ell$-limited-magnitude and $s$ adjacent transpositions.

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arXiv2023

Codes for Correcting $t$ Limited-Magnitude Sticky Deletions

Codes for correcting sticky insertions/deletions and limited-magnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$-sticky deletions with $\ell$-limited-magnitude and propose a non-systematic code for correcting this type of error with redundancy $2t(1-1/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\frac{\lceil2t(1-1/p)\rceil\cdot\lceil\log p\rceil}{\log p} \log(n+1)+O(\log\log n)$, where $p$ is the smallest prime larger than $\ell+1$.

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arXiv2021

Endurance-Limited Memories: Capacity and Codes

\emph{Resistive memories}, such as \emph{phase change memories} and \emph{resistive random access memories} have attracted significant attention in recent years due to their better scalability, speed, rewritability, and yet non-volatility. However, their \emph{limited endurance} is still a major drawback that has to be improved before they can be widely adapted in large-scale systems. In this work, in order to reduce the wear out of the cells, we propose a new coding scheme, called \emph{endurance-limited memories} (\emph{ELM}) codes, that increases the endurance of these memories by limiting the number of cell programming operations. Namely, an \emph{$\ell$-change $t$-write ELM code} is a coding scheme that allows to write $t$ messages into some $n$ binary cells while guaranteeing that each cell is programmed at most $\ell$ times. In case $\ell=1$, these codes coincide with the well-studied \emph{write-once memory} (\emph{WOM}) codes. We study some models of these codes which depend upon whether the encoder knows on each write the number of times each cell was programmed, knows only the memory state, or even does not know anything. For the decoder, we consider these similar three cases. We fully characterize the capacity regions and the maximum sum-rates of three models where the encoder knows on each write the number of times each cell was programmed. In particular, it is shown that in these models the maximum sum-rate is $\log \sum_{i=0}^{\ell} {t \choose i}$. We also study and expose the capacity regions of the models where the decoder is informed with the number of times each cell was programmed. Finally we present the most practical model where the encoder read the memory before encoding new data and the decoder has no information about the previous states of the memory.

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arXiv2020

Constrained de Bruijn Codes: Properties, Enumeration, Constructions, and Applications

The de Bruijn graph, its sequences, and their various generalizations, have found many applications in information theory, including many new ones in the last decade. In this paper, motivated by a coding problem for emerging memory technologies, a set of sequences which generalize sequences in the de Bruijn graph are defined. These sequences can be also defined and viewed as constrained sequences. Hence, they will be called constrained de Bruijn sequences and a set of such sequences will be called a constrained de Bruijn code. Several properties and alternative definitions for such codes are examined and they are analyzed as generalized sequences in the de Bruijn graph (and its generalization) and as constrained sequences. Various enumeration techniques are used to compute the total number of sequences for any given set of parameters. A construction method of such codes from the theory of shift-register sequences is proposed. Finally, we show how these constrained de Bruijn sequences and codes can be applied in constructions of codes for correcting synchronization errors in the $\ell$-symbol read channel and in the racetrack memory channel. For this purpose, these codes are superior in their size on previously known codes.

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