Factoring Groups into Subsets

Szabo, Sylvia; Sands, A. D.

doi:10.1201/9781420090475

Cited by 59 publications

(62 citation statements)

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“…Splittings were first considered in [19] in connection with the problem of tiling Euclidean space by translates of certain polytopes composed of unit cubes, called k-crosses and k-semicrosses, see also [9] and [20,21,23,24,25]. Perfect splitter sets are equivalent to codes correcting single limited magnitude errors in flash memories (see [1], [5,10,11,12,13,16,17,27,29,30,31] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On the existence of perfect splitter sets

Yuan

Zhao

2020

Finite Fields and Their Applications

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Given integers k 1 , k 2 with 0 ≤ k 1 < k 2 , the determinations of all positive integers q for which there exists a perfect Splitter B[−k 1 , k 2 ](q) set is a wide open question in general. In this paper, we obtain new necessary and sufficient conditions for an odd prime p such that there exists a nonsingular perfect B[−1, 3](p) set. We also give some necessary conditions for the existence of purely singular perfect splitter sets. In particular, we determine all perfect B[−k 1 , k 2 ](2 n ) sets for any positive integers k 1 , k 2 with k 1 + k 2 ≥ 4. We also prove that there are infinitely many prime p such that there exists a perfect B[−1, 3](p) set.

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Section: Introductionmentioning

confidence: 99%

On the existence of perfect splitter sets

Yuan

Zhao

2020

Finite Fields and Their Applications

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“…The study of factorizations of abelian groups was initiated by G. Hajós [12] in his proof of a well-known conjecture of Minkowski. Since then there has been extensive research on factorizations and tilings of abelian groups; see for example [9,26,27] for related results and background information. As far as we know, in the literature most studies on tilings are about the abelian case.…”

Section: Introductionmentioning

confidence: 99%

Perfect Codes in Cayley Graphs

Huang¹,

Xia²,

Zhou³

2018

SIAM J. Discrete Math.

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Given a graph Γ, a subset C of V (Γ) is called a perfect code in Γ if every vertex of Γ is at distance no more than one to exactly one vertex in C, and a subset C of V (Γ) is called a total perfect code in Γ if every vertex of Γ is adjacent to exactly one vertex in C. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions.

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“…Following [18], we say that B is a complementer factor to A and G = AB is a factorization of G. In this case, we have the partitions…”

Section: Kaleidoscopical Configurations In Groupsmentioning

confidence: 99%

“…The factorizations were introduced by Hajoś [5] to solve the famous Minkowsky's problem on tiling of R n by the copies of a cube. For modern state of factorizations see [17,18]. Also we establish a connection between kaleidoscopical configurations and T -sequences from [12].…”

Section: Introductionmentioning

confidence: 99%

Kaleidoscopical configurations

Протасов

Protasova

2014

J Math Sci

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Abstract.Let G be a group and X be a G-space with the action G × X → X, (g, x) → gx. A subset A of X is called a kaleidoscopical configuration if there is a coloring χ : X → κ (i.e. a mapping of X onto a cardinal κ) such that the restriction χ|gA is a bijection for each g ∈ G. We survey some recent results on kaleidoscopical configurations in metric spaces considered as G-spaces with respect to the groups of its isometries and in groups considered as left regular G-spaces.2010 MSC. 05A18, 05B30.

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Factoring Groups into Subsets

Cited by 59 publications

References 0 publications

On the existence of perfect splitter sets

On the existence of perfect splitter sets

Perfect Codes in Cayley Graphs

Kaleidoscopical configurations

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