Lilya Budaghyan, Claude Carlet, Alexander Pott
We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.
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In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications; for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist.
Multidimensional convolution matrices and perfect colorings of subspace hypergraphs applied for bent functions and related designs
本文通过多维卷积矩阵与子空间超图的完美着色理论，研究了与弯曲函数相关的组合设计及其应用。
Combinatorial designs, difference sets and bent functions as perfect colorings of graphs and multigraphs
本文建立独立集、超图横截集与完美着色的等价关系，并证明组合设计、$q$-设计、差集、Hadamard 矩阵及 bent 函数可表示为特定图或多重图的完美着色。
Drawing inspiration from Nyberg's paper~\cite{Nyb91} on perfect nonlinearity and the $c$-differential notion we defined in~\cite{EFRST20}, in this paper we introduce the concept of $c$-differential bent functions in two different ways (thus extending Kumar et al.~\cite{Ku85} classical definition). We further extend the notion of perfect $c$-nonlinear introduced in~\cite{EFRST20}, also in two different ways, and show that, in both cases, the concepts of $c$-differential bent and perfect $c$-nonlinear are equivalent (under some natural restriction of the parameters). Some constructions of functi…
Almost Maiorana-McFarland bent functions
本文研究了广义Maiorana-McFarland类中的几乎Maiorana-McFarland bent函数，刻画其bent性质、确定对偶，提出构造算法生成${M}^\n#$外函数，解决开放问题并实现类内外函数的相互生成。
Some observations on bent and planar functions
本文研究了弯曲函数和平面函数的性质，证明了其图的Salem集性质，并展示了平面函数间距离的下限为二。
Relative Difference sets from Almost Perfect Nonlinear Functions
本文研究了几乎完美非线性函数与相对差集之间的关系，并通过Pot的结果揭示了APN函数与弯曲函数的联系。
In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in $n$ variables for all $n\ge 10$. Moreover, we show that for almost all values of $n$, these functions can simultaneously be homogeneous and have no affine derivatives.
Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In this paper, we study monomial GAPN functions for odd characteristic. In particular, we give all monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic.
Let $T$ be a lobster with a matching that covers all but one vertex. We show that in this case, $T$ is graceful.
For an $m$-cycle $C$, an inside $m$-cycle of $C$ is a cycle on the same vertex set, that is edge-disjoint from $C$. In an $m$-cycle system, $(\mathcal{X}, \mathcal{C})$, if inside $m$-cycles can be chosen -one for each cycle- to form another $m$-cycle system, then $(\mathcal{X}, \mathcal{C})$ is called an almost $2$-perfect $m$-cycle system. Almost $2$-perfect cycle systems can be considered as generalisations of $2$-perfect cycle systems. Cycle packings are generalisations of cycle systems that allow to have leaves after decomposition. In this paper, we prove that an almost $2$-perfect maximu…
The minimum co-degree threshold for a perfect matching in a $k$-graph with $n$ vertices was determined by Rödl, Ruciński and Szemerédi for the case when $n\equiv 0\pmod k$. Recently, Han resolved the remaining cases when $n \not\equiv 0\pmod k$, establishing a conjecture of Rödl, Ruciński and Szemerédi. In this paper, we determine the minimum co-degree threshold for almost perfect matchings in $k$-partite $k$-graphs, answering a question of Rödl and Ruciński.
Trees maximizing the number of almost-perfect matchings
本文刻画了最大化近乎完美匹配与强近乎完美匹配数量、并最小化极大匹配数量的极值树，并将结果用于加权 Hosoya 指数的极值问题。
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for exampl…
It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved this conjecture for linear Lee codes with $r=2$. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius $2$. We show that if such codes exist in $\mathbb{Z}^n$, then $n$ must be $1,2, 11, 29, 47, 56, 67, 79, 104, 121, 134$ or $191$.
We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number $α(P_n)$ and clique number $ω(P_n)$ is close to a concentrated distribution $L(n)$ which plays an important role in our generation method. We also prove that the probability that $P_n$ contains any give…
Almost perfect graph classes
本文基于强完美图定理，刻画了|H|≤2时不含H诱导子图的图为c-顶点可删完美图的情况，并将结果扩展到完美图的多个子类。
More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved that if $C$ is linear, then the Golomb-Welch conjecture is valid for $r=2$ and $n\geq 3$. In this paper, we consider the classification of linear Lee codes with the second-best possibility, that is the density of the lattice packing of $\mathbb{Z}^n$ by Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. We show that, for $r=2$ and $n\equiv 0,3,4 \pmod{6}$, this packing density can n…
The absolute values of the perfect matching derangement graph's eigenvalues almost follow the lexicographic order of partitions
本研究通过推导K_{2n}的完美匹配错位排列图ℳ_{2n}特征值的新递推公式，证明了其特征值绝对值存在与普通错位排列图类似的、近似遵循分划字典序的性质，拓展了错位排列图特征值序的相关结论。
Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonli…
Almost-perfect colorful matchings in three-edge-colored bipartite graphs
该研究证明了三边着色二分图中存在满足各颜色边数指定条件的几乎完美彩色匹配，验证了Ryser-Brualdi-Stein猜想的三色多重扩展版本，且所得和的上界为最优。
For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whethe…
A uniform bound on almost colour-balanced perfect matchings in colour-balanced cliques
研究色平衡团中寻找尽可能接近色平衡的完美匹配问题，通过消除对$n$的依赖，将最小化颜色偏差的上界从$\mathcal{O}(k\sqrt{nk\log k})$改进为一致界$4^{k^2}$。
A stability result for almost perfect matchings
本文研究满足特定参数条件的非平凡$k$-图的最大边数问题，确定了顶点数为$n$、匹配数至多为$s$的这类图的边数上界，证实了Frankl提出的相关猜想在$s$充分大时的正确性。
Extremal Matchings and Height Functions
本文研究圆盘上平面二分图（plabic图）中几乎完美匹配的格结构，通过高度函数显式构造极值匹配，并证明所有边界条件均可由此获得。
A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is `pseudorandom', meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.
Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings
该研究证明了Glock等人关于高围长$(n,q,r)$-Steiner系统存在性猜想的近似版本，推广了超图匹配相关经典定理，具有多项应用价值。
In 1965 Erdős asked, what is the largest size of a family of $k$-element subsets of an $n$-element set that does not have a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^{3}$ and $(s+1)k\le n<(s+1)(k+\frac{1}{100k})$.
On de Bruijn Rings and Families of Almost Perfect Maps
本文提出了一种构造任意形状和非质数字母大小的近完美映射的方法，通过引入德布劳恩环，解决了部分德布劳恩托里存在的难题，并适用于位置编码应用。
The concept of differential uniformity was recently extended to the $c$-differential uniformity. An interesting problem in this area is the construction of functions with low $c$-differential uniformity and a lot of research has been done in this direction in the recent past. Here, we present three classes of (almost) perfect $c$-nonlinear permutations over finite fields of even characteristic.
Triprojective almost perfect nonlinear permutations and functions
本文提出了一类新的三投影结构诱导的几乎完美非线性排列和函数，解决了有限向量空间中奇数维度可被三整除时的构造问题，并给出了非双射的高非线性函数。
For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomassé [Almost all H-free graphs have the Erdős-Hajnal property] in the case of forbidden i…
Hamilton paths and cycles in flip graphs of (almost-)perfect matchings
研究图匹配间的翻转操作及对应翻转图的哈密顿性。组合情形下证明翻转图是哈密顿连通或哈密顿可穿的；几何情形下证明路径会错过指数多个顶点，无哈密顿路径或圈，还为凸点集和几乎完美匹配构造了几乎遍历所有顶点的圈。
Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low $c$-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results.
Almost-perfect packings and Tuza's conjecture in the random geometric graph
研究随机几何图中Tuza猜想成立的条件，并证明对于一大类密度该猜想成立；同时研究用固定图F的边不交副本覆盖几乎所有边的问题，对无限族F证明了几乎完美打包的存在性。
Changing almost perfect nonlinear functions on affine subspaces of small codimensions
针对 APN 函数，本文研究其代数分解与二次构造，给出修改后仍为 APN 的精确判别准则，并证明所得函数可与原函数扩展仿射不等价。
Generalized Almost Perfect Nonlinear Binomials and Trinomials Over Fields of Prime-Square Order
该研究针对素数$p>3$的$\mathbb{F}_{p^2}$域，推导GAPN二项式的充分条件，显式构造出$p$到$2(p-1)$次的奇次GAPN二项式，结合已有刻画构造出对应范围的偶次GAPN三项式，填补了奇特征扩域偶次GAPN函数的空白。
In a prior paper [14], along with P. Ellingsen, P. Felke and A. Tkachenko, we defined a new (output) multiplicative differential, and the corresponding c-differential uniformity, which has the potential of extending differential cryptanalysis. Here, we continue the work, by looking at some APN functions through the mentioned concept and show that their c-differential uniformity increases significantly, in some cases.
Value Distributions of Perfect Nonlinear Functions
本文利用纯组合工具开发一般性框架研究完全非线性函数的值分布，推导出基本阿贝尔情形下的新条件，并完全确定了输出维数至多为4的向量布尔bent函数的值分布，同时证明了值分布在判定函数完全非线性性与等价性方面的作用。
Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$ with $n$ vertices, in every step $i \geq 1$ the graph $G_i$ is formed by taking an edge chosen uniformly at random among the non-existing ones and adding it to the graph $G_{i - 1}$. The classical `hitting-time' result of Ajtai, Komlós, and Szemerédi, and independently Bollobás, states that asymptotically almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches $2$, that is if $δ(G_i) \ge 2$ then $G_i$ is Hamiltonian. We establish a resilience version of this result. In particular, we show th…
Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $\frac{φ(m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil$ inequivalent APN functions on the finite field with ${2^{2m}}$ elements, where $φ$ denotes Euler's totient function. This is a great improvement of previous results: for even $m$, the best known lower …
In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these coefficients. To our knowledge this provides the best known bound on both the PTF density (number of monomials) and weight (sum of the coefficient magnitudes) of general Boolean functions. The special case of Bent functions is also analyzed and shown that any n-variable Bent function can be represented with integer coefficients less than $2^n$ while also obeying…
We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain [JS89] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least $Ω(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.
Equivalences of biprojective almost perfect nonlinear functions
针对几乎完全非线性函数的计数与等价性问题，本文提出强群论方法，解决了所有已知双射影APN函数的等价性问题，提出新函数族并证明其可产生指数级新不等价函数，为该类函数研究提供通用方法。
Almost perfect nonlinear power functions with exponents expressed as fractions
本文研究了一类APN幂函数的指数表达形式，通过重新表述指数形式，实现了对差分谱的精确分析，并确定了任意差分导数纤维中的元素。
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