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Register a new userNew nonexistence results on $(m,n)$-generalized bent functions
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Added by
Qi Wang
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, we present some new nonexistence results on $(m,n)$-generalized bent functions, which improved recent results. More precisely, we derive new nonexistence results for general $n$ and $m$ odd or $m equiv 2 pmod{4}$, and further explicitly prove nonexistence of $(m,3)$-generalized bent functions for all integers $m$ odd or $m equiv 2 pmod{4}$. The main tools we utilized are certain exponents of minimal vanishing sums from applying characters to group ring equations that characterize $(m,n)$-generalized bent functions.
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This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahns information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but it may be irregular in the other, the extended entropy-based proof technique yields the same bound that was conjectured by Kahn (2001) and proved by Sah et al. (2019).
We prove the nonexistence of lattice tilings of $mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $ngeq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a prime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound.
In this paper, we show that, for all $ngeq 5$, the maximum number of $2$-chains in a butterfly-free family in the $n$-dimensional Boolean lattice is $leftlceilfrac{n}{2}rightrceilbinom{n}{lfloor n/2rfloor}$. In addition, for the height-2 poset $K_{s,t}$, we show that, for fixed $s$ and $t$, a $K_{s,t}$-free family in the $n$-dimensional Boolean lattice has $Oleft(nbinom{n}{lfloor n/2rfloor}right)$ $2$-chains.
Let f be a function mapping an n dimensional vector space over GF(p) to GF(p). When p is 2, Bernasconi et al. have shown that there is a correspondence between certain properties of f (e.g., if it is bent) and properties of its associated Cayley graph. Analogously, but much earlier, Dillon showed that f is bent if and only if the level curves of f had certain combinatorial properties (again, only when p is 2). The attempt is to investigate an analogous theory when p is greater than 2 using the (apparently new) combinatorial concept of a weighted partial difference set. More precisely, we try to investigate which properties of the Cayley graph of f can be characterized in terms of function-theoretic properties of f, and which function-theoretic properties of f correspond to combinatorial properties of the set of level curves, i.e., the inverse map of f. While the natural generalizations of the Bernasconi correspondence and Dillon correspondence are not true in general, using extensive computations, we are able to determine a classification in some small cases. Our main conjecture is Conjecture 67.
Generalized Turan problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants.
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