arXiv subjects

Gennian Ge

Publications and source records attributed to Gennian Ge.

At least 55 records · Page 3Linked to original sources

arXiv2025

Reconstruction Codes for Deletions and Insertions: Connection, Distinction, and Construction

Let $\mathcal{B}(\cdot)$ be an error ball function. A set of $q$-ary sequences of length $n$ is referred to as an \emph{$(n,q,N;\mathcal{B})$-reconstruction code} if each sequence $\boldsymbol{x}$ within this set can be uniquely reconstructed from any $N$ distinct elements within its error ball $\mathcal{B}(\boldsymbol{x})$. The main objective in this area is to determine or establish bounds for the minimum redundancy of $(n,q,N;\mathcal{B})$-reconstruction codes, denoted by $ρ(n,q,N;\mathcal{B})$. In this paper, we investigate reconstruction codes where the error ball is either the \emph{$t$-deletion ball} $\mathcal{D}_t(\cdot)$ or the \emph{$t$-insertion ball} $\mathcal{I}_t(\cdot)$. Firstly, we establish a fundamental connection between reconstruction codes for deletions and insertions. For any positive integers $n,t,q,N$, any $(n,q,N;\mathcal{I}_t)$-reconstruction code is also an $(n,q,N;\mathcal{D}_t)$-reconstruction code. This leads to the inequality $ρ(n,q,N;\mathcal{D}_t)\leq ρ(n,q,N;\mathcal{I}_t)$. Then, we identify a significant distinction between reconstruction codes for deletions and insertions when $N=O(n^{t-1})$ and $t\geq 2$. For deletions, we prove that $ρ(n,q,\tfrac{2(q-1)^{t-1}}{q^{t-1}(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{D}_t)=O(1)$, which disproves a conjecture posed in \cite{Chrisnata-22-IT}. For insertions, we show that $ρ(n,q,\tfrac{(q-1)^{t-1}}{(t-1)!}n^{t-1}+O(n^{t-2});\mathcal{I}_t)=\log\log n + O(1)$, which extends a key result from \cite{Ye-23-IT}. Finally, we construct $(n,q,N;\mathcal{B})$-reconstruction codes, where $\mathcal{B}\in \{\mathcal{D}_2,\mathcal{I}_2\}$, for $N \in \{2,3, 4, 5\}$ and establish respective upper bounds of $3\log n+O(\log\log n)$, $3\log n+O(1)$, $2\log n+O(\log\log n)$ and $\log n+O(\log\log n)$ on the minimum redundancy $ρ(n,q,N;\mathcal{B})$. This generalizes results previously established in \cite{Sun-23-IT}.

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arXiv2025

All rectangles exhibit canonical Ramsey property

In a seminal work, Cheng and Xu proved that for any positive integer $r$, there exists an integer $n_0$, independent of $r$, such that every $r$-coloring of the $n$-dimensional Euclidean space $\mathbb{E}^n$ with $n \ge n_0$ contains either a monochromatic or a rainbow congruent copy of a square. This phenomenon of dimension-independence was later formalized as the canonical Ramsey property by Geheér, Sagdeev, and Tóth, who extended the result to all hypercubes, and to rectangles whose side lengths $a$, $b$ satisfy $(\frac{a}{b})^2$ is rational. They further posed the natural problem of whether every rectangle admits the canonical Ramsey property, regardless of the aspect ratio. In this paper, we show that all rectangles exhibit the canonical Ramsey property, thereby completely resolving this open problem of Geheér, Sagdeev, and Tóth. Our proof introduces a new structural reduction that identifies product configurations with bounded color complexity, enabling the application of simplex Ramsey theorems and product Ramsey amplification to control arbitrary aspect ratios.

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arXiv2025

On the Existence and Nonexistence of Splitter Sets

In this paper, the existence of perfect and quasi-perfect splitter sets in finite abelian groups is studied, motivated by their application in coding theory for flash memory storage. For perfect splitter sets we view them as splittings of $\mathbb{Z}_n$, and using cyclotomic polynomials we derive a general condition for the existence of such splittings under certain circumstances. We further establish a relation between $B[-k, k](q)$ and $B[-(k-1), k+1](q)$ splitter sets, and give a necessary and sufficient condition for the existence of perfect $B[-1, 5](q)$ splitter sets. Finally, two nonexistence results for quasi-perfect splitter sets are presented.

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arXiv2025

Correcting Bursty/Localized Deletions: A New Error-Position-Estimation Code

Codes correcting bursts of deletions and localized deletions have garnered significant research interest in recent years. One of the primary objectives is to construct codes with minimal redundancy. Currently, the best known constructions of $q$-ary codes correcting a burst of at most $t$ deletions ($(\le t)$-burst-deletion correcting codes) achieve redundancy $\log n+8\log\log n+o(\log\log n)$ (for any $q$ and $t$) or $\log n+t\log\log n+O(1)$ (for even $q$). For codes correcting single $t$-localized-deletion ($t$-localized-deletion correcting codes), state-of-the-art constructions attain redundancy $\log n+O\parenv{t(\log\log n)^2}$ (for any $q$ and $t$) or $\log n+2t\log\log n+O(1)$ (for even $q$). Here, $n$ denotes the code-length, and $q$ and $t$ are fixed. These codes employ a position-estimation component to approximate error positions, augmented by additional constraints that enable error-correction given the information about error positions. In this work, we select codewords from the set of sequences whose differential sequences are strong-$(\ell,ε)$-locally-balanced. By imposing a VT-type constraint and an $L_1$-weight constraint on the differential sequences of codewords, we construct novel position-estimation codes. When $q\ge 2$ and $t<q$, or $q$ is even and $t<2q$, this approach gives a $q$-ary $(\le t)$-burst-deletion correcting code and a $t$-localized-deletion correcting code with redundancy $\log n+(t-1)\log\log n+O(1)$. In addition to improving previous redundancy, the method is new and our position-estimation codes are simpler than those in previous works. Finally, we give an efficient encoder to encode an arbitrary input sequence into a sequence whose differential sequence is strong-$(\ell,ε)$-locally-balanced. To our knowledge, no prior algorithm for this specific task has been reported.

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arXiv2025

On the Maximum Size of Codes Under the Damerau-Levenshtein Metric

The Damerau-Levenshtein distance between two sequences is the minimum number of operations (deletions, insertions, substitutions, and adjacent transpositions) required to convert one sequence into another. Notwithstanding a long history of this metric, research on error-correcting codes under this distance has remained limited. Recently, motivated by applications in DNA-based storage systems, Gabrys \textit{et al} and Wang \texit{et al} reinvigorated interest in this metric. In their works, some codes correcting both deletions and adjacent transpositions were constructed. However, theoretical upper bounds on code sizes under this metric have not yet been established. This paper seeks to establish upper bounds for code sizes in the Damerau-Levenshtein metric. Our results show that the code correcting one deletion and asymmetric adjacent transpositions proposed by Wang \textit{et al} achieves optimal redundancy up to an additive constant.

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arXiv2025

Optimal Reconstruction Codes with Given Reads in Multiple Burst-Substitutions Channels

We study optimal reconstruction codes over the multiple-burst substitution channel. Our main contribution is establishing a trade-off between the error-correction capability of the code, the number of reads used in the reconstruction process, and the decoding list size. We show that over a channel that introduces at most $t$ bursts, we can use a length-$n$ code capable of correcting $ε$ errors, with $Θ(n^ρ)$ reads, and decoding with a list of size $O(n^λ)$, where $t-1=ε+ρ+λ$. In the process of proving this, we establish sharp asymptotic bounds on the size of error balls in the burst metric. More precisely, we prove a Johnson-type lower bound via Kahn's Theorem on large matchings in hypergraphs, and an upper bound via a novel variant of Kleitman's Theorem under the burst metric, which might be of independent interest. Beyond this main trade-off, we derive several related results using a variety of combinatorial techniques. In particular, along with tools from recent advances in discrete geometry, we improve the classical Gilbert-Varshamov bound in the asymptotic regime for multiple bursts, and determine the minimum redundancy required for reconstruction codes with polynomially many reads. We also propose an efficient list-reconstruction algorithm that achieves the above guarantees, based on a majority-with-threshold decoding scheme.

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arXiv2025

Correcting Errors Through Partitioning and Burst-Deletion Correction

In this paper, we propose a partitioning technique that decomposes a pair of sequences with overlapping $t$-deletion $s$-substitution balls into sub-pairs, where the $^{\leq}t$-burst-deletion balls of each sub-pair intersect. This decomposition facilitates the development of $t$-deletion $s$-substitution correcting codes that leverage approaches from $^{\leq}t$-burst-deletion correction. Building upon established approaches in the $^{\leq}t$-burst-deletion correction domain, we construct $t$-deletion $s$-substitution correcting codes for $t\in \{1,2\}$ over binary alphabets and for $t=1$ in non-binary alphabets, with some constructions matching existing results and others outperforming current methods. Our framework offers new insights into the underlying principles of prior works, elucidates the limitations of current approaches, and provides a unified perspective on error correction strategies.

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arXiv2025

On the Fixed-Length-Burst Levenshtein Ball with Unit Radius

Consider a length-$n$ sequence $\bm{x}$ over a $q$-ary alphabet. The \emph{fixed-length Levenshtein ball} $\mathcal{L}_t(\bm{x})$ of radius $t$ encompasses all length-$n$ $q$-ary sequences that can be derived from $\bm{x}$ by performing $t$ deletions followed by $t$ insertions. Analyzing the size and structure of these balls presents significant challenges in combinatorial coding theory. Recent studies have successfully characterized fixed-length Levenshtein balls in the context of a single deletion and a single insertion. These works have derived explicit formulas for various key metrics, including the exact size of the balls, extremal bounds (minimum and maximum sizes), as well as expected sizes and their concentration properties. However, the general case involving an arbitrary number of $t$ deletions and $t$ insertions $(t>1)$ remains largely uninvestigated. This work systematically examines fixed-length Levenshtein balls with multiple deletions and insertions, focusing specifically on \emph{fixed-length burst Levenshtein balls}, where deletions occur consecutively, as do insertions. We provide comprehensive solutions for explicit cardinality formulas, extremal bounds (minimum and maximum sizes), expected size, and concentration properties surrounding the expected value.

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arXiv2025

New Wide Locally Recoverable Codes with Unified Locality

Wide Locally Recoverable Codes (LRCs) have recently been proposed as a solution for achieving high reliability, good performance, and ultra-low storage cost in distributed storage systems. However, existing wide LRCs struggle to balance optimal fault tolerance and high availability during frequent system events. By analyzing the existing LRCs, we reveal three limitations in the LRC construction which lay behind the non-optimal overall performance from multiple perspectives, including non-minimum local recovery cost, non cluster-topology-aware data distribution, and non XOR-based local coding. Thanks to the flexible design space offered by the locality property of wide LRCs, we present UniLRC, which unifies locality considerations in code construction. UniLRC achieves the optimal fault tolerance while overcoming the revealed limitations. We implement UniLRC prototype and conduct comprehensive theoretical and system evaluations, showing significant improvements in reliability and performance over existing wide LRCs deployed in Google and Azure clusters.

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arXiv2025

Binary Reconstruction Codes for Correcting One Deletion and One Substitution

In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function $ \mathcal{B} $ as a mapping from a sequence to the set of sequences that can be derived from it by performing one deletion and one substitution. A binary \emph{$(n,N;\mathcal{B})$-reconstruction code} is defined as a collection of binary sequences of length $ n $ such that the intersection size between the single-deletion single-substitution balls of any two distinct codewords is strictly less than $ N $. This property ensures that each codeword can be uniquely reconstructed from $ N $ distinct elements in its single-deletion single-substitution ball. Our main contribution is to demonstrate that when $ N $ is set to $ 4n - 8 $, $ 3n - 4 $, $2n+9$, $ n+21 $, $31$, and $7$, the redundancy of binary $(n,N;\mathcal{B})$-reconstruction codes can be $0$, $1$, $2$, $ \log\log n + 3 $, $\log n + 1 $, and $ 3\log n + 4 $, respectively, where the logarithm is on base two.

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arXiv2025

Sequence Reconstruction under Channels with Multiple Bursts of Insertions or Deletions

The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence. Levenshtein established that this number exceeds the maximum size of the intersection between the error balls of any two distinct transmitted sequences by one. In this paper, we consider channels subject to multiple bursts of insertions and multiple bursts of deletions, respectively, where each burst has an exact length of value b. We provide a complete solution for the insertion case while partially addressing the deletion case.

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arXiv2025

Optimal redundancy of function-correcting codes

Function-correcting codes, introduced by Lenz, Bitar, Wachter-Zeh, and Yaakobi, protect specific function values of a message rather than the entire message. A central challenge is determining the optimal redundancy -- the minimum additional information required to recover function values amid errors. This redundancy depends on both the number of correctable errors $t$ and the structure of message vectors yielding identical function values. While prior works established bounds, key questions remain, such as the optimal redundancy for functions like Hamming weight and Hamming weight distribution, along with efficient code constructions. In this paper, we make the following contributions: (1) For the Hamming weight function, we improve the lower bound on optimal redundancy from $\frac{10(t-1)}{3}$ to $4t - \frac{4}{3}\sqrt{6t+2} + 2$. On the other hand, we provide a systematical approach to constructing explicit FCCs via a novel connection with Gray codes, which also improve the previous upper bound from $\frac{4t-2}{1 - 2\sqrt{\log{2t}/(2t)}}$ to $4t - \log{t}$. Consequently, we almost determine the optimal redundancy for Hamming weight function. (2) The Hamming weight distribution function is defined by the value of Hamming weight divided by a given positive integer $T$. Previous work established that the optimal redundancy is $2t$ when $T > 2t$, while the case $T \le 2t$ remained unclear. We show that the optimal redundancy remains $2t$ when $T \ge t+1$. However, in the surprising regime where $T = o(t)$, we achieve near-optimal redundancy of $4t - o(t)$. Our results reveal a significant distinction in behavior of redundancy for distinct choices of $T$.

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arXiv2025

Asymptotically Optimal Codes for $(t,s)$-Burst Error

Recently, codes for correcting a burst of errors have attracted significant attention. One of the most important reasons is that bursts of errors occur in certain emerging techniques, such as DNA storage. In this paper, we investigate a type of error, called a $(t,s)$-burst, which deletes $t$ consecutive symbols and inserts $s$ arbitrary symbols at the same coordinate. Note that a $(t,s)$-burst error can be seen as a generalization of a burst of insertions ($t=0$), a burst of deletions ($s=0$), and a burst of substitutions ($t=s$). Our main contribution is to give explicit constructions of $q$-ary $(t,s)$-burst correcting codes with $\log n + O(1)$ bits of redundancy for any given constant non-negative integers $t$, $s$, and $q \geq 2$. These codes have optimal redundancy up to an additive constant. Furthermore, we apply our $(t,s)$-burst correcting codes to combat other various types of errors and improve the corresponding results. In particular, one of our byproducts is a permutation code capable of correcting a burst of $t$ stable deletions with $\log n + O(1)$ bits of redundancy, which is optimal up to an additive constant.

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arXiv2024

Algebraic approach to stability results for Erdős-Ko-Rado theorem

Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erdős, Ko, and Rado in 1961 established that any $k$-uniform intersecting family on $[n]$ has a maximum size of $\binom{n-1}{k-1}$, with the unique extremal structure being a star. In 1967, Hilton and Milner followed up with a pivotal result, showing that if such a family is not a star, its size is at most $\binom{n-1}{k-1} - \binom{n-k-1}{k-1} + 1$, and they identified the corresponding extremal structures. In recent years, Han and Kohayakawa, Kostochka and Mubayi, and Huang and Peng have provided the second and third levels of stability results in this line of research. In this paper, we provide a unified approach to proving the stability result for the Erdős-Ko-Rado theorem at any level. Our framework primarily relies on a robust linear algebra method, which leverages appropriate non-shadows to effectively handle the structural complexities of these intersecting families.

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arXiv2024

Linear Network Coding for Robust Function Computation and Its Applications in Distributed Computing

We investigate linear network coding in the context of robust function computation, where a sink node is tasked with computing a target function of messages generated at multiple source nodes. In a previous work, a new distance measure was introduced to evaluate the error tolerance of a linear network code for function computation, along with a Singleton-like bound for this distance. In this paper, we first present a minimum distance decoder for these linear network codes. We then focus on the sum function and the identity function, showing that in any directed acyclic network there are two classes of linear network codes for these target functions, respectively, that attain the Singleton-like bound. Additionally, we explore the application of these codes in distributed computing and design a distributed gradient coding scheme in a heterogeneous setting, optimizing the trade-off between straggler tolerance, computation cost, and communication cost. This scheme can also defend against Byzantine attacks.

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arXiv2024

A polynomial resultant approach to algebraic constructions of extremal graphs

The Turán problem asks for the largest number of edges ex$(n,H)$ in an $n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one of the most important problems in extremal graph theory. However the order of magnitude of ex$(n,H)$ for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomail resultant approach to algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lowers bounds for Turán number of $1$-subdivision of $K_{3,t_{1}}$ and linear Turán number of Berge theta hyerpgraph $Θ_{3,t_{2}}^{B}$ with $t_{1}=25$ and $t_{2}=217$. Moreover, the constant $t_{1}$ improves the random algebraic construction of Bukh and Conlon~[Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the known estimation for the smallest value of $t_{1}$ concerning a problem posed by Conlon, Janzer and Lee ~[More on the extremal number of subdivisions, Combinatorica, to appear], while the constant $t_{2}$ improves a result of He and Tait~[Hypergraphs with few berge paths of fixed length between vertices, SIAM J. Discrete Math., 33(3), 1472-1481].

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arXiv2024

Constrained coding upper bounds via Goulden-Jackson cluster theorem

Motivated by applications in DNA-based data storage, constrained codes have attracted a considerable amount of attention from both academia and industry. We study the maximum cardinality of constrained codes for which the constraints can be characterized by a set of forbidden substrings, where by a substring we mean some consecutive coordinates in a string. For finite-type constrained codes (for which the set of forbidden substrings is finite), one can compute their capacity (code rate) by the ``spectral method'', i.e., by applying the Perron-Frobenious theorem to the de Brujin graph defined by the code. However, there was no systematic method to compute the exact cardinality of these codes. We show that there is a surprisingly powerful method arising from enumerative combinatorics, which is based on the Goulden-Jackson cluster theorem (previously not known to the coding community), that can be used to compute not only the capacity, but also the exact formula for the cardinality of these codes, for each fixed code length. Moreover, this can be done by solving a system of linear equations of size equal to the number of constraints. We also show that the spectral method and the cluster method are inherently related by establishing a direct connection between the spectral radius of the de Brujin graph used in the first method and the convergence radius of the generating function used in the second method. Lastly, to demonstrate the flexibility of the new method, we use it to give an explicit upper bound on the maximum cardinality of variable-length non-overlapping codes, which are a class of constrained codes defined by an infinite number of forbidden substrings.

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arXiv2024

Binary Codes for Correcting Two Edits

An edit refers to a single insertion, deletion, or substitution. This paper aims to construct binary codes that can correct two edits. To do this, a necessary and sufficient condition for a code to be two-edit correctable is provided, showing that a code is a two-edit correcting code if and only if it can correct two deletions, up to two substitutions, and one deletion and up to one substitution, separately. This criterion allows for the construction of two-edit correcting codes leveraging these three types of error correcting codes. In the field of constructing codes for correcting two deletions, we present a construction with $4\log n+O(\log\log n)$ redundant bits that can be viewed as a subcode proposed by Guruswami and Håstad, and provide an alternative proof. Moreover, our two-deletion correcting codes can also correct up to two substitutions after making a slight modification. In the field of constructing codes for correcting one deletion and up to one substitution, we present a construction with $4 \log n+O(\log\log n)$ redundant bits, which outperforms the best previously known results $6 \log n+O(1)$. Leveraging these codes, we obtain a construction of two-edit correcting codes with $6 \log n+O(\log\log n)$ redundant bits. This outperforms the best previously known result, which requires at least $8\log n$ redundant bits. Moreover, we also consider the list-decoding problem under the two-edit channel and construct a two-edit list-decodable code with a list size of two employing $4 \log n+O(\log\log n)$ redundant bits.

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