arXiv subjects

Gennian Ge

Publications and source records attributed to Gennian Ge.

At least 19 recordsLinked to original sources

arXiv2026

On the weighted hard-core model and Rado's covering problem for congruent Euclidean balls

Let $K$ be a symmetric convex body in $\mathbb{R}^d$ and let $f(K)$ denote the largest constant $c$ such that every finite collection of translates of $K$ contains a pairwise disjoint subcollection whose total volume is at least $c$ times the volume of the union of the original collection. The classical Vitali covering lemma gives $f(K)\geq3^{-d}$. In this paper, we establish two improvements. First, by a purely combinatorial argument, we prove that $$ f(K)\geq \frac{2}{3^d + 2^d} $$ for every symmetric convex body $K$. This improves the Vitali bound by a factor tending to $2$ as $d$ tends to infinity. Second, using a weighted hard-core model together with a weighted geometric estimate for intersections of Euclidean balls, we show that, for all sufficiently large $d$, $$ f(B^d)\geq \left( \log\frac{3}{1+\sqrt3} -O\left(\frac{\log d}{d}\right) \right)d\,3^{-d}, $$ where $B^d$ is the unit Euclidean ball in $\mathbb{R}^d$. Thus, the classical lower bound is improved by a factor of order $d$.

math.MG↗

arXiv2026

Economical lattice coverings by determined segments

For fixed $d\geq 2$, let $\tau_d(n)$ be the minimum size of a set $S\subseteq\{0,\ldots,n\}^d$ such that the affine lines determined by pairs of distinct points of $S$ cover the grid. Let $\sigma_d(n)$ be the analogous minimum when every grid point must lie on the closed segment joining two distinct points of $S$. A celebrated result of Alon [GAFA, 1991] proved that $\tau_d(n)$ is of order between $\Omega_d(n^{\alpha_d})$ and $O_d(n^{\alpha_d}\log n)$, where $\alpha_d=\frac{d(d-1)}{2d-1}$, and asked whether the logarithm term is necessary. We prove that $$c_d n^{\alpha_d}\leq\tau_d(n)\leq\sigma_d(n)\leq C_d n^{\alpha_d}$$ for every fixed $d\geq 2$, thereby resolving Alon's problem in a stronger form.

math.CO↗

arXiv2026

Linear network codes for vector-linear network function computation over three-layer networks

We study vector-linear function computation over three-layer networks with a fixed target function and a fixed source-access pattern. We develop a support-constrained row-space framework that represents a linear computing code by a global row space. This space must contain the target row space and be generated by rows satisfying the local support-constraints of the network. We prove that this representation is equivalent to the existence of a linear computing code. For any prescribed global row space, we give a necessary and sufficient condition for its realization and determine the minimum uniform communication load at the middle nodes. The condition is expressed in terms of the ranks of the local subspaces supported on the source-access sets. It separates the exact local realization problem from the outer problem of designing the global row space and yields a variational characterization of the linear computing capacity. We then apply the framework to MDS targets over cyclic networks. We identify when the target row space alone is sufficient and when auxiliary rows are required. We determine the capacity in the dense regime and in the sparse regime whenever the cut-set bound is integral. For the remaining sparse parameters, we give a general linear construction whose achievable rate equals the integer part of the cut-set bound.

cs.IT↗

arXiv2026

Polynomially Improved Lower Bounds for Trifferent Codes via Locally Sparse $3$-Uniform Hypergraphs

A ternary code is \emph{trifferent} if every three distinct codewords have a coordinate in which their symbols are pairwise distinct. Let $T(n)$ be the maximum size of a trifferent code of length $n$. The classical K\"orner--Marton construction gives $T(n)\ge c_0(9/5)^{n/4}$ for an absolute constant $c_0>0$. We prove the polynomial strengthening $T(n)\ge c\sqrt{n}(9/5)^{n/4}$ for an absolute constant $c>0$. Our proof refines the outer-code step in the K\"orner--Marton concatenation. We encode non separating triples as edges of a $3$-uniform hypergraph, randomly thin its vertex set, and remove high-degree vertices together with all remaining Berge cycles of lengths two and three. The resulting locally sparse hypergraph admits a large independent set by a theorem of Verstraete and Wilson, producing the additional factor $\sqrt n$. Concatenation with the length-four Tetra code then yields the stated lower bound.

cs.IT↗

arXiv2026

Largest Sperner families with restricted differences

Let $L$ be a fixed set of positive integers. A family $\mathcal{F}\subseteq 2^{[n]}$ is called $L$-differencing if $\lvert A\setminus B\rvert\in L$ for every ordered pair of distinct members $A,B\in\mathcal{F}$. A longstanding conjecture of Frankl, proposed in 1985, asserts that every $L$-differencing family has size at most $\binom{n}{|L|}$. We resolve this conjecture asymptotically for every fixed $L$, and obtain the exact answer in the only case in which the conjectured bound could be tight. (1) If $L\ne [s]$ and $n$ is large, then every $L$-differencing family satisfies $\lvert \mathcal{F}\rvert \le \left(\frac{s}{s+1}+o_L(1)\right)\binom{n}{s}$. (2) If $L=[s]$ and $n\ge 2s-1$, then $\lvert \mathcal{F}\rvert\le\binom{n}{s}$, with equality only for $\binom{[n]}{s}$ and $\binom{[n]}{n-s}$. The first result follows by reducing directed differences to restricted Hamming distances. For the exact result, we develop a new homogeneous polynomial method, which might be of independent interest.

math.CO↗

arXiv2026

Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions

Let $f_{r}(d,s_{1},\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\subseteq\mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\sqcup\cdots\sqcup P_{r}$ with the following property: whenever $C_{i}\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\bigcap_{i=1}^{r}C_{i}\ne\emptyset$. A recent breakthrough of Alon and Smorodinsky proved that $f_{r}(d,s,\ldots,s)\le cdr^{2}s^{r}\log r\log(es^{r})$ for an absolute constant $c>0$. In this paper, we determine the asymptotic order in two principal ranges: $f_{2}(2,s,s)=\Theta(s^{2})$, and $f_{r}(d,s,\ldots,s)=\Theta_{d,r}(s^{r}\log{s})$ for every fixed $d,r$ with $d\ge r+2$. The first one determines the order of the extremal function proposed by Kalai from the 1970s. Together, the two results show a sharp dependence on the dimension: for two parts, the logarithmic factor disappears in the plane but is necessary in every fixed dimension $d\ge4$. Beyond these sharp results, when $r\ge d+1$ we improve the upper bound of Alon and Smorodinsky by proving both $f_{r}(d,s,\ldots,s)\le c_{d}rs^{r}\log(ers^{r})$ and $f_{r}(d,s,\ldots,s)\le c_{d}r^{d+2}s^{d+1}\log(ers)$ through a local Helly-type argument. We also prove $f_{r}(d,s,\ldots,s)>s^{\min\{r,d\}}$ for every $d\ge2$. Finally, we study two colored analogues. The direct B\'{a}r\'{a}ny--Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, a different extension does hold: given sufficiently many prescribed $r$-point classes, one can split every class completely among the $r$ final parts while retaining the required intersection property.

math.CO↗

arXiv2026

All simplices exhibit canonical Ramsey property

We prove that all nondegenerate simplices have the canonical Ramsey property, thereby resolving a central open problem in canonical Euclidean Ramsey theory and providing a canonical counterpart to the celebrated simplex Ramsey theorem of Frankl and R\"{o}dl~[JAMS, 1990].

math.CO↗

arXiv2026

Recursive Lifting Beyond the Ahlswede--Khachatrian Construction

For the Erd\H{o}s--Frankl--Pach problem on uniform set systems of bounded VC-dimension, the Ahlswede--Khachatrian/Mubayi--Zhao construction has long served as the standard lower-bound benchmark. We develop a recursive lifting method that goes beyond this benchmark in every dimension $d\ge3$, proving that for every $d\ge3$ and $n\ge d+3$, \[ M_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+M_{d-3}(n-5). \] The proof is elementary and proceeds through explicit trace obstructions. We also record a further recursive improvement in the concluding remarks.

math.CO↗

arXiv2026

New Results on Limited Magnitude Error Correcting Codes

This paper investigates the existence, construction and classification of limited magnitude error-correcting codes, with a focus on splitter sets and their connections to group splittings. We establish new nonexistence results for quasi-perfect splitter sets and provide a complete classification of quasi-perfect $B[0,3](n)$ splitter sets in both singular and nonsingular cases. Furthermore, we derive improved lower bounds for the size of maximal $B[0,3](q)$ sets by investigating Cayley graphs, where $q$ is a prime. We also provide existence criteria for perfect $B[0,6](q)$ splitter sets and quasi-perfect $B[-4,4](2p)$ sets for prime $p$. For perfect burst-correcting codes, we develop a general construction framework, and prove the existence of infinite families of $(k_2,k_1)$-limited-magnitude cyclic $b$-burst-correcting codes for $k_1+k_2\le 4$ and arbitrary burst length $b$. We further provide sufficient existence conditions for general parameters $k_1$ and $k_2$. Our results combine algebraic, combinatorial, and number-theoretic methods to advance the understanding of codes tailored for flash memory and related storage systems.

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arXiv2026

Extremal Deletion-Ball Intersections under Run-Count and Lower-Order Deletion-Ball Intersection Constraints

Motivated by sequence reconstruction and reconstruction codes, we study extremal intersections of deletion balls over a fixed $q$-ary alphabet. Let $\Sigma_q^n$ be the set of sequences of length $n$ over $\Sigma_q$, and let $D_t(x)$ denote the set of all sequences obtained from $x\in\Sigma_q^n$ by deleting exactly $t$ symbols. Our first result gives a finite upper bound under a lower-order deletion-correction constraint. We prove that if $x,y\in\Sigma_q^n$ satisfy $D_{s-1}(x)\cap D_{s-1}(y)=\varnothing$, then \[ |D_t(x)\cap D_t(y)| \le \binom{2s}{s}\binom{n-s}{t-s}. \] For binary alphabets, this strengthens a recent asymptotic upper bound of Pham, Goyal, and Kiah (2025, JCTA). We then investigate deletion-ball intersections under simultaneous constraints on run counts and lower-order deletion-ball intersections. For fixed $0<\gamma\le1$, integers $1\le s\le t$, and $m\ge1$, we show that if $x,y\in\Sigma_q^n$ have at most $\gamma n$ runs and satisfy $|D_s(x)\cap D_s(y)|\le m$, then \[ |D_t(x)\cap D_t(y)|\le \frac{m\gamma^{t-s}}{(t-s)!}n^{t-s}+O_{s,t,m}(n^{t-s-1}). \] Moreover, the leading term can be attainable whenever $m$ is realized by a fixed finite-length seed pair. As a consequence, we obtain a direct lifting theorem for deletion reconstruction codes, transferring reconstruction properties from radius $s$ to larger radii $t$. Finally, we establish a parallel insertion theory and derive corresponding results for insertion-ball intersections and insertion reconstruction codes.

math.CO↗

arXiv2026

Deletion-Correcting Codes for the $\ell$-Symbol Read Channel

This paper studies deletion-correcting codes for the $\ell$-symbol read channel, whose noiseless output is the vector of all consecutive $\ell$-mers of a transmitted sequence. This model is motivated by overlapping-read mechanisms arising in nanopore sequencing, racetrack memories with consecutive read heads, and related sequence-labeling problems. We consider an adversarial setting in which a fixed number of $\ell$-mers are deleted from the read vector. Our first contribution is a structural characterization of the effect of such deletions: after a minimum number of $\ell$-mers are inserted to restore consistency, the resulting sequence is obtained from the transmitted sequence by deleting symbols from certain periodic substrings; when $t\le \ell-2$, these deletions correspond to complete minimum periods. Based on this characterization, we introduce check patterns and construct $\ell$-read deletion-correcting codes via power-sum syndromes. For every $\ell\ge2$, we obtain single-deletion correcting codes with redundancy $\log\lfloor (n+2\ell)/(\ell-1)\rfloor$. For $2\le t\le \ell/2$, we construct $q$-ary $\ell$-read $t$-deletion correcting codes with redundancy $t\log n+O(1)$, and for $\ell=2t-1$ with $t\ge3$, we construct codes with redundancy $(2t-1)\log n+O(1)$. We also study the sporadic parameter pairs $(\ell,t)\in\{(2,2),(3,2),(3,3)\}$ and obtain improved constructions, including binary $\ell$-read $2$-deletion correcting codes with redundancy $2\log n+O(1)$ for $\ell=2,3$, a non-binary $3$-read $2$-deletion correcting code with redundancy $3\log n+O(1)$, a binary $3$-read $3$-deletion correcting code with redundancy $5\log n+O(1)$, and a non-binary $3$-read $3$-deletion correcting code with redundancy $7\log n+O(1)$.

cs.IT↗

arXiv2026

Random Reed--Solomon Codes Correcting Permutations, Insertions, and Deletions over Polynomial-Size Alphabets

We study Reed--Solomon codes against adversarial coordinate permutations followed by insertion-deletion (insdel) errors. It was previously shown by Con (2025) that Reed--Solomon codes can attain the exact half-Singleton bound in this setting, but only over exponentially large alphabets. We prove that, by allowing an additive $\epsilon n$ gap from this bound, the alphabet size can be reduced to polynomial. More precisely, for fixed constants $R,\epsilon\in(0,1)$ satisfying $2R+\epsilon<1$ and $k=Rn$, a random Reed--Solomon code of length $n$ and dimension $k$ over an alphabet of size $n^{O_{R,\epsilon}(1)}$ is, with high probability, robust against arbitrary coordinate permutations followed by up to $(1-\epsilon)n-2k+1$ insdel errors. We also prove a complementary alphabet-size lower bound, showing that positive-rate codes, which are robust against linearly many insdel errors in the permutation-insdel setting, require a polynomially superlinear alphabet. Finally, for the explicit two-dimensional Reed--Solomon codes constructed by Con et al. (2024) over alphabet size $O(n^3)$, we give an average $O(n)$-time decoder against arbitrary coordinate permutations followed by $n-3$ insdel errors. Previously, an $O(n)$-time decoder for this code was known only for the deletion setting.

cs.IT↗

arXiv2026

Recursive lower bounds for uniform set systems of bounded VC-dimension

For integers $n\ge d+1$, let $\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}. \] We introduce a two-cover lifting construction and prove the recursive lower bound \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+\mathsf{M}_{d-3}(n-5) \] for every $d\ge 3$ and $n\ge d+3$. Consequently, \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+\binom{n-6}{d-3}. \] Thus the Mubayi--Zhao conjecture on the exact value of $\mathsf{M}_d(n)$ for $n\ge2(d+2)$ is false for any $d\ge 3$. The proof is elementary and proceeds entirely through an explicit analysis of traces.

math.CO↗

arXiv2026

New bounds for covering codes under insertions or deletions

Covering codes for insertions and deletions arise naturally in the study of synchronization errors and differ substantially from their classical counterparts in the Hamming metric. In this paper, we study covering codes under insertion and deletion operations. We first show that, in contrast to the equivalence between insertion and deletion correction, insertion covering and deletion covering are not equivalent. We then develop bounds and constructions for insertion and deletion covering codes, with particular emphasis on the large-alphabet regime. For insertion covering codes, we extend a recent combinatorial approach for single insertions and establish a new lower bound for arbitrary fixed insertion radius. For deletion covering codes, we relate the problem to hypergraph covering and prove that the elementary counting lower bound is asymptotically tight when the alphabet size tends to infinity. We further provide a construction of asymptotically optimal non-binary single-deletion covering codes by using differential Varshamov--Tenengolts (VT) codes together with a completion argument. In addition, we study covering codes for burst deletions. We prove that binary differential VT codes are not only capable of correcting two-burst deletions but also have the corresponding covering property, and hence form binary perfect codes for two-burst deletions. Finally, we extend this construction to non-binary alphabets and obtain explicit $q$-ary two-burst-deletion covering codes.

cs.IT↗

arXiv2026

Universal Fourier-inertia bounds for prescribed even distances

The study of set families with restricted Hamming distances is a classical topic of extremal combinatorics and coding theory. Let $H=\{A\subseteq[n]: |A|\text{ is even}\}$ be the even subcube. Let $\ell_1,\dots,\ell_t$ be distinct positive integers and set $\mathcal L=\{2\ell_1,\dots,2\ell_t\}$. We prove that, for all sufficiently large $n$, every family $\mathcal F\subseteq H$ satisfying $ |A\triangle B|\in\mathcal L $ for all $A\ne B\in\mathcal F$ has \[ |\mathcal F|\le \sum_{i=0}^{t}\binom{n-1}{i}. \] This is best possible as a universal bound, with equality attained at the distance set $\mathcal L=\{2,4,\dots,2t\}$. Our proof uses a Fourier-inertia argument based on a universal low/high boundary-layer sign pattern for the Fourier transform of the distance-polynomial kernel on the even subcube: the prescribed distances enter only through lower-order Fourier terms, while the leading boundary-layer signs depend solely on $t$. This even-subcube result immediately yields an odd-subcube analogue and, through parity reductions, provides bounds for arbitrary distance sets. In particular, this approach recovers the classical interval bounds of Kleitman and the corresponding interval bounds of Huang--Klurman--Pohoata, while offering a direct spectral proof of the maximality of $\{2,4,\dots,2t\}$ among all fixed $t$-distance sets.

math.CO↗

arXiv2026

Monochromatic unit equilateral triangle on low-dimensional spheres

A result of Matou\v{s}ek and R\"odl in 1995 states that for every $\varepsilon>0$ and every triangle $T$ with circumradius $\rho(T)$, there exists a dimension $n=n(\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $\rho(T)+\varepsilon$, namely $\mathbb{S}^{n}(\rho(T)+\varepsilon)$, contains a monochromatic congruent copy of $T$. In this paper, we determine the exact threshold dimension for the unit equilateral triangle on the sphere $\mathbb{S}^{n}(1/\sqrt{2})$: there exists a $2$-coloring of $\mathbb{S}^{2}(1/\sqrt{2})$ with no monochromatic unit equilateral triangle, whereas every $2$-coloring of $\mathbb{S}^{3}(1/\sqrt{2})$ contains one. Along the way, we also establish several further Euclidean Ramsey-type results on low-dimensional spheres, including asymmetric and isosceles variants.

math.CO↗

arXiv2026

Correcting One Deletion and One Substitution with a Constant Number of Reads

In this paper, we investigate the problem of designing $(n, N; \mathcal{B})$-reconstruction codes for $N\in \{14,11,9,5\}$, where $\mathcal{B}$ is the single-deletion single-substitution ball function that maps a sequence to the set of all sequences obtainable via one deletion and one substitution. Such a code is defined by the requirement that the intersection size of any two distinct single-deletion single-substitution balls is strictly less than the given number of noisy reads $N$. Note that for any $1\le N<N'$, an $(n, N; \mathcal{B})$-reconstruction code is also an $(n, N'; \mathcal{B})$-reconstruction code. It follows that the problem of designing $(n, N; \mathcal{B})$-reconstruction codes with less redundancy becomes more challenging as $N$ decreases, particularly because the problem for $N=1$ already reduces to the coding problem of single-deletion and single-substitution correcting codes. To the best of our knowledge, most existing results focus on the case where $N$ is a linear function of $n$, while only a limited number consider constant $N$. When $N=1$, the best known $(n, 1; \mathcal{B})$-reconstruction codes (single-deletion and single-substitution correcting codes) require $(4+o(1))\log n$ redundant bits. In this work, we show that this redundancy can be reduced to $3\log n+4$ when $N=5$. As $N$ increases further to $9$ and $11$, the redundancy can be improved to $2\log n+12\log\log n+O(1)$ and $\log n +12\log \log n+O(1)$, respectively. Finally, for $N=14$, we provide a reconstruction code with $\log n+3$ bits of redundancy, which is only two bits more than the best known $(n, 18; \mathcal{B})$-reconstruction codes.

cs.IT↗

arXiv2026

Improvement on the Erd\H{o}s-Kleitman conjecture via the KKL theorem

In 1974, Erd\H{o}s and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size $s$ and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For decades, the best general lower bound remained the trivial $2^{n-1}$. About a decade ago, Frankl and Tokushige emphasized that obtaining a bound of the form $\left(\frac{1}{2}+\varepsilon\right)\cdot 2^n$ for some $\varepsilon>0$ is a challenging problem. A breakthrough of Buci\v{c}, Letzter, Sudakov and Tran in 2018 showed that $ |\mathcal{F}|\ge \left(1-\frac{1}{s}\right)\cdot 2^n $ via two very elegant and quite different approaches. Our main result shows that $$ |\mathcal{F}|\ge \left( 1 - \frac{1}{s + (s-2)\frac{\log n}{2\sqrt{5}n}} \right)\cdot 2^n $$ by exploiting a connection to the cornerstone result of Kahn, Kalai and Linial on influences of Boolean functions. Independently, we can also obtain a weaker improvement combining the linear algebra method with a combinatorial twist.

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