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Register a new userNo lattice tiling of $mathbb{Z}^n$ by Lee Sphere of radius 2
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Yue Zhou
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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.
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Let $tgeq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $muleft( Gright) $ satisfies [ muleft( Gright) leqfrac{t-1}{2}+sqrt{n+frac{t^{2}-2t-3}{4}}, ] with equality if and only if $nequiv1$ $(operatorname{mod}$ $t)$ and $G=K_{1}veeleftlfloor n/trightrfloor K_{t}$. For $t=3$ the maximum $muleft( Gright) $ is found exactly for any $n>40000$.
Hefetz, M{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $Gamma$ be an Abelian group of order $n$. A textit{directed $Gamma$-distance magic labeling} of an oriented graph $vec{G} = (V,A)$ of order $n$ is a bijection $vec{l}:V rightarrow Gamma$ with the property that there is a textit{magic constant} $mu in Gamma$ such that for every $x in V(G)$ $ w(x) = sum_{y in N^{+}(x)}vec{l}(y) - sum_{y in N^{-}(x)} vec{l}(y) = mu. $ In this paper we provide an infinite family of odd regular graphs possessing an orientable $mathbb{Z}_{n}$-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable $mathbb{Z}_{n}$-distance magic.
Physical platforms such as trapped ions suffer from coherent noise where errors manifest as rotations about a particular axis and can accumulate over time. We investigate passive mitigation through decoherence free subspaces, requiring the noise to preserve the code space of a stabilizer code, and to act as the logical identity operator on the protected information. Thus, we develop necessary and sufficient conditions for all transversal $Z$-rotations to preserve the code space of a stabilizer code, which require the weight-$2$ $Z$-stabilizers to cover all the qubits that are in the support of some $X$-component. Further, the weight-$2$ $Z$-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the size of these components, we are able to construct a large family of QECC codes, oblivious to coherent noise, that includes the $[[4L^2, 1, 2L]]$ Shor codes. Moreover, given $M$ even and any $[[n,k,d]]$ stabilizer code, we can construct an $[[Mn, k, ge d]]$ stabilizer code that is oblivious to coherent noise. If we require that transversal $Z$-rotations preserve the code space only up to some finite level $l$ in the Clifford hierarchy, then we can construct higher level gates necessary for universal quantum computation. The $Z$-stabilizers supported on each non-zero $X$-component form a classical binary code C, which is required to contain a self-dual code, and the classical Gleasons theorem constrains its weight enumerator. The conditions for a stabilizer code being preserved by transversal $2pi/2^l$ $Z$-rotations at $4 le l le l_{max} <infty$ level in the Clifford hierarchy lead to generalizations of Gleasons theorem that may be of independent interest to classical coding theorists.
For each natural number $d$, we introduce the concept of a $d$-cap in $mathbb{F}_3^n$. A subset of $mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in $mathbb{F}_3^n$. We prove that the $2$-caps in $mathbb{F}_3^n$ are exactly the Sidon sets in $mathbb{F}_3^n$ and study the problem of determining the size of the largest $2$-cap in $mathbb{F}_3^n$.
The odd wheel $W_{2k+1}$ is the graph formed by joining a vertex to a cycle of length $2k$. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an $n$-vertex graph that does not contain $W_{2k+1}$. We determine the structure of the spectral extremal graphs for all $kgeq 2, k otin {4,5}$. When $k=2$, we show that these spectral extremal graphs are among the Tur{a}n-extremal graphs on $n$ vertices that do not contain $W_{2k+1}$ and have the maximum number of edges, but when $kgeq 9$, we show that the family of spectral extremal graphs and the family of Tur{a}n-extremal graphs are disjoint.
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