arXiv subjects

Xiande Zhang

Publications and source records attributed to Xiande Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Counting cliques with prescribed intersection sizes

We study the generalized Turán problem regarding cliques with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subset [0,r-1]$ be a fixed integer set with $|L|\notin \{1,r\}$ and $\ell_1<\dots<\ell_s$, and let $Ψ_r(n,L)$ denote the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques are $L$-intersecting as a family of $r$-subsets. Helliar and Liu recently initiated the systematic study of the function $Ψ_r(n,L)$ and showed that $Ψ_r(n,L)\le \left(1-\frac{1}{3r}\right) \prod_{\ell\in L}\frac{n-\ell}{r-\ell}$ for large $n$, improving the trivial bound from the Deza--Erdős--Frankl theorem by a factor of $1-\frac{1}{3r}$. In this article, we improve their result by showing that as $n$ goes to infinity $Ψ_r(n,L)=Θ_{r,L}(n^{|L|})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and fully determining the corresponding exact values of $Ψ_r(n,L)$ for sufficiently large $n$ in this case. Moreover, when $L=[t,r-1]$, for the generalized Turán extension of the Erdős--Ko--Rado theorem given by Helliar and Liu, we show a Hilton--Milner-type stability result.

math.CO↗

arXiv2025

Linear Programming Bounds on $k$-Uniform States

The existence of $k$-uniform states has been a widely studied problem due to their applications in several quantum information tasks and their close relation to combinatorial objects like Latin squares and orthogonal arrays. With the machinery of quantum enumerators and linear programming, we establish several improved non-existence results and bounds on $k$-uniform states. 1. First, for any fixed $l\geq 1$ and $q\geq 2$, we show that there exists a constant $c$ such that $(\left\lfloor{n/2}\right\rfloor-l)$-uniform states in $(\mathbb{C}^q)^{\otimes n}$ do not exist when $n\geq cq^2+o(q^2)$. The constant $c$ equals $4$ when $l=1$ and $6$ when $l=2$, which generalizes Scott's bound (2004) for $l=0$. 2. Second, when $n$ is sufficiently large, we show that there exists a constant $θ<1/2$ for each $q \le 9$, such that $k$-uniform states in $(\mathbb{C}^q)^{\otimes n}$ exist only when $k\leq θn$. In particular, this provides the first bound (to the best of our knowledge) of $k$ for $4\leq q\leq 9$ and confirms a conjecture posed by Shi et al. (2023) when $q=5$ in a stronger form. 3. Finally, we improve the shadow bounds given by Shi et al. (2023) by a constant for $q = 3,4,5$ and small $n$. When $q=4$, our results can update some bounds listed in the code tables maintained by Grassl (2007--2024).

quant-ph↗

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 $T$.

cs.IT↗

arXiv2024

On low-power error-correcting cooling codes with large distances

A low-power error-correcting cooling (LPECC) code was introduced as a coding scheme for communication over a bus by Chee et al. to control the peak temperature, the average power consumption of on-chip buses, and error-correction for the transmitted information, simultaneously. Specifically, an $(n, t, w, e)$-LPECC code is a coding scheme over $n$ wires that avoids state transitions on the $t$ hottest wires and allows at most $w$ state transitions in each transmission, and can correct up to $e$ transmission errors. In this paper, we study the maximum possible size of an $(n, t, w, e)$-LPECC code, denoted by $C(n,t,w,e)$. When $w=e+2$ is large, we establish a general upper bound $C(n,t,w,w-2)\leq \lfloor \binom{n+1}{2}/\binom{w+t}{2}\rfloor$; when $w=e+2=3$, we prove $C(n,t,3,1) \leq \lfloor \frac{n(n+1)}{6(t+1)}\rfloor$. Both bounds are tight for large $n$ satisfying some divisibility conditions. Previously, tight bounds were known only for $w=e+2=3,4$ and $t\leq 2$. In general, when $w=e+d$ is large for a constant $d$, we determine the asymptotic value of $C(n,t,w,w-d)\sim \binom{n}{d}/\binom{w+t}{d}$ as $n$ goes to infinity, which can be extended to $q$-ary codes.

cs.IT↗

arXiv2024

Extremal Maximal Entanglement

A pure multipartite quantum state is called absolutely maximally entangled if all reductions of no more than half of the parties are maximally mixed. However, an $n$-qubit absolutely maximally entangled state only exists when $n$ equals $2$, $3$, $5$, and $6$. A natural question arises when it does not exist: which $n$-qubit pure state has the largest number of maximally mixed $\lfloor n/2 \rfloor$-party reductions? Denote this number by $Qex(n)$. It was shown that $Qex(4)=4$ in [Higuchi et al.Phys. Lett. A (2000)] and $Qex(7)=32$ in [Huber et al.Phys. Rev. Lett. (2017)]. In this paper, we give a general upper bound of $Qex(n)$ by linking the well-known Turán's problem in graph theory, and provide lower bounds by constructive and probabilistic methods. In particular, we show that $Qex(8)=56$, which is the third known value for this problem.

quant-ph↗

arXiv2024

Improvements on Permutation Reconstruction from Minors

We study the reconstruction problem of permutation sequences from their $k$-minors, which are subsequences of length $k$ with entries renumbered by $1,2,\ldots,k$ preserving order. We prove that the minimum number $k$ such that any permutation of length $n$ can be reconstructed from the multiset of its $k$-minors is between $\exp{(Ω(\sqrt{\ln n}))}$ and $O(\sqrt{n\ln n})$. These results imply better bounds of a well-studied parameter $N_d$, which is the smallest number such that any permutation of length $n\ge N_d$ can be reconstructed by its $(n-d)$-minors. The new bounds are $ d+\exp(Ω(\sqrt{\ln d}))<N_d<d+O(\sqrt{d\ln d})$ asymptotically, and the previous bounds were $d+\log_2 d<N_d<d^2/4+2d+4$.

math.CO↗

arXiv2024

Focal-free uniform hypergraphs and codes

Motivated by the study of a variant of sunflowers, Alon and Holzman recently introduced focal-free hypergraphs. In this paper, we show that there is an interesting connection between the maximum size of focal-free hypergraphs and the renowned Erdős Matching Conjecture on the maximum number of edges that can be contained in a uniform hypergraph with bounded matching number. As a consequence, we give asymptotically optimal bounds on the maximum sizes of focal-free uniform hypergraphs and codes, thereby significantly improving the previous results of Alon and Holzman. Moreover, by using the existentce results of combinatorial designs and orthogonal arrays, we are able to explicitly determine the exact sizes of maximum focal-free uniform hypergraphs and codes for a wide range of parameters.

math.CO↗

arXiv2024

Trace reconstruction of matrices and hypermatrices

A \emph{trace} of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied \emph{trace reconstruction} problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multivariate version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy et al. showed that $\exp(\widetilde{O}(n^{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$) and any unknown $n^{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result, we improve this upper bound by showing that $\exp(\widetilde{O}(n^{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp(\widetilde{O}(n^{3/5}))$ traces suffice to reconstruct any unknown $n^{\times d}$ hypermatrix. This breaks the tendency to trivial $\exp(O(n))$ as the dimension $d$ grows.

math.CO↗

arXiv2024

Unextendible product operator basis

Quantum nonlocality is associated with the local indistinguishability of orthogonal states. Unextendible product basis (UPB), a widely used tool in quantum information, exhibits nonlocality which is the powerful resource for quantum information processing. In this work we extend the definitions of nonlocality and genuine nonlocality from states to operators. We also extend UPB to the notions of unextendible product operator basis, unextendible product unitary operator basis (UPUOB) and strongly UPUOB. We construct their examples, and show the nonlocality of some strongly UPUOBs under local operations and classical communications. We study the phenomenon of these operators acting on quantum states. As an application, we distinguish the two-dimensional strongly UPUOB which only consumes three ebits of entanglement. Our results imply that such UPUOBs exhibit nonlocality as UPBs and the distinguishability of them requires entanglement resources.

quant-ph↗

arXiv2024

Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$

A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the $N$-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in $N$-partite systems for all odd $N\geq 3$. Based on this decomposition, we give explicit constructions of unextendible product bases in $N$-partite systems for odd $N\geq 3$. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.

quant-ph↗

arXiv2024

On set systems with strongly restricted intersections

Set systems with strongly restricted intersections, called $α$-intersecting families for a vector $α$, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector $α=(a_1, \ldots, a_k)$, a collection $\mathcal F$ of subsets over an $n$ element set is an $α$-intersecting family modulo $2$ if for each $i=1,2,\ldots,k$, all $i$-wise intersections of distinct members in $\mathcal F$ have sizes with the same parity as $a_i$. Let $f_α(n)$ denote the maximum size of such a family. In this paper, we study the asymptotic behavior of $f_α(n)$ when $n$ goes to infinity. We show that if $t$ is the maximum integer such that $a_t=1$ and $2t\leq k$, then $f_{α(n)} \sim {(t! n)}^{\frac 1 t}$. More importantly, we show that for any constant $c$, as the length $k$ goes larger, $f_α(n)$ is upper bounded by $O (n^c)$ for almost all $α$. Equivalently, no matter what $k$ is, there are only finitely many $α$ satisfying $f_α(n)=Ω(n^c)$. This answers an open problem raised by Johnston and O'Neill in 2023. All of our results can be generalized to modulo $p$ setting for any prime $p$ smoothly.

math.CO↗

arXiv2024

Improved upper bounds for wide-sense frameproof codes

Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. In this paper, we investigate upper bounds on the cardinality of wide-sense $t$-frameproof codes. For $t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For $t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length $n$ is at most $\frac{15+\sqrt{33}}{24}(t-1)^2$, we show that a wide-sense $t$-frameproof code has at most $n$ codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve several best known results on binary $t$-frameproof codes.

math.CO↗

arXiv2024

Almost all even-particle pure states are determined by their half-body marginals

Determining whether the original global state is uniquely determined by its local marginals is a prerequisite for some efficient tools for characterizing quantum states. This paper shows that almost all generic pure states of even $N$-particle with equal local dimension are uniquely determined among all other pure states (UDP) by four of their half-body marginals. Furthermore, we give a graphical description of the marginals for determining genuinely multipartite entangled states, which leads to several lower bounds on the number of required marginals. Finally, we present a construction of N-qudit states obtained from certain combinatorial structures that cannot be UDP by its k-body marginals for some k>N/2-1.

quant-ph↗

arXiv2024

Reconstruction of hypermatrices from subhypermatrices

For a given $n$, what is the smallest number $k$ such that every sequence of length $n$ is determined by the multiset of all its $k$-subsequences? This is called the $k$-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case -- reconstruction of $n\times n$-matrices from submatrices. Previous works show that the smallest $k$ is at most $O(n^\frac{1}{2})$ for sequences and at most $O(n^\frac{2}{3})$ for matrices. We study this $k$-deck problem for general dimension $d$ and prove that, the smallest $k$ is at most $O(n^\frac{d}{d+1})$ for reconstructing a $d$ dimensional hypermatrix of order $n$ from the multiset of all its subhypermatrices of order $k$.

math.CO↗

arXiv2023

Bounds on $k$-Uniform Quantum States

Do $N$-partite $k$-uniform states always exist when $k\leq \lfloor\frac{N}{2}\rfloor-1$? In this work, we provide new upper bounds on the parameter $k$ for the existence of $k$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ when $d=3,4,5$, which extend Rains' bound in 1999 and improve Scott's bound in 2004. Since a $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ corresponds to a pure $((N,1,k+1))_{d}$ quantum error-correcting codes, we also give new upper bounds on the minimum distance $k+1$ of pure $((N,1,k+1))_d$ quantum error-correcting codes. Furthermore, we generalize Scott's bound to heterogeneous systems, and show some non-existence results of absolutely maximally entangled states in $\mathbb{C}^{d_1}\otimes(\mathbb{C}^{d_2})^{\otimes 2n}$.

quant-ph↗

arXiv2023

Swap-Robust and Almost Supermagic Complete Graphs for Dynamical Distributed Storage

To prevent service time bottlenecks in distributed storage systems, the access balancing problem has been studied by designing almost supermagic edge labelings of certain graphs to balance the access requests to different servers. In this paper, we introduce the concept of robustness of edge labelings under limited-magnitude swaps, which is important for studying the dynamical access balancing problem with respect to changes in data popularity. We provide upper and lower bounds on the robustness ratio for complete graphs with $n$ vertices, and construct $O(n)$-almost supermagic labelings that are asymptotically optimal in terms of the robustness ratio.

math.CO↗

arXiv2023

Separating hash families with large universe

Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of universe for a $t$-perfect hash family of length $N$ over an alphabet of size $q$. In this paper, we show that $q^{2-o(1)}<p_t(t, q)=o(q^2)$ for all $t\geq 3$, which answers an open problem about separating hash families raised by Blackburn et al. in 2008 for certain parameters. Previously, this result was known only for $t=3, 4$. Our proof is obtained by establishing the existence of a large set of integers avoiding nontrivial solutions to a set of correlated linear equations.

math.CO↗

arXiv2023

On supersaturation for oddtown and eventown

We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show that if $\mathcal A$ consists of $n+s$ odd-sized subsets, then $op(\mathcal A)\geq s+2$, which is tight when $s\le n-4$. This disproves a conjecture by O'Neill on the supersaturation problem of oddtown. For the supersaturation problem of eventown, we show that for large enough $n$, if $\mathcal A$ consists of $2^{\lfloor \frac n 2\rfloor}+s$ even-sized subsets, then $op(\mathcal A)\ge s\cdot2^{\lfloor \frac n 2\rfloor-1} $ for any positive integer $s\le \frac{2^{\lfloor\frac n 8\rfloor}} n$. This partially proves a conjecture by O'Neill on the supersaturation problem of eventown. Previously, the correctness of this conjecture was only verified for $s=1$ and $2$. We further provide a twice weaker lower bound in this conjecture for eventown, that is $op(\mathcal{A})\ge s\cdot 2^{\lfloor n/2\rfloor-2}$ for general $n$ and $s$ by using discrete Fourier analysis. Finally, some asymptotic results for the lower bounds of $op(\mathcal A)$ are given when $s$ is large for both problems.

math.CO↗