Weighted sums in finite cyclic groups

Gao, Weidong; Jin, Xin

doi:10.1016/j.disc.2004.02.001

Cited by 13 publications

(4 citation statements)

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“…index(G) was first introduced by Chapman, Freeze and Smith in [2]. In [6], Gao obtained the bounds as: n+1 2 + 1 l(G) n − n+1 3 + 1 for all n 8, and l(G) = 1 for n = 1, 2, 3, 4, 5, 7, l(G) = 5 for n = 6. We prove that l(G) = n 2 + 2 for n 8.…”

Section: Introductionmentioning

confidence: 98%

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On the index of minimal zero-sum sequences over finite cyclic groups

Yuan

2007

Journal of Combinatorial Theory, Series A

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Let G be a cyclic group of order n 2 and S = g 1 · · · · · g k a sequence over G. We say that S is a zero-sum sequence if k i=1 g i = 0 and that S is a minimal zero-sum sequence if S is a zero-sum sequence and S contains no proper zero-sum sequence.The notion of the index of a minimal zero-sum sequence (see Definition 1.1) in G has been recently addressed in the mathematical literature. Let l(G) be the smallest integer t ∈ N such that every minimal zero-sum sequence S over G with length |S| t satisfies index(S) = 1. In this paper, we first prove that l(G) = n 2 + 2 for n 8. Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences.

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Section: Introductionmentioning

confidence: 98%

“…Let l(G) denote the smallest integer t ∈ N such that every minimal zero-sum sequence S ∈ F(G) with length |S| t satisfies index(S) = 1. (For the existence of t, we refer to [6] and the references given there. )…”

Section: Introductionmentioning

confidence: 99%

On the index of minimal zero-sum sequences over finite cyclic groups

Yuan

2007

Journal of Combinatorial Theory, Series A

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“…of zerosum free sequences) over cyclic groups. It was first addressed by Lemke-Kleitman (in the conjecture [8, page 344]), used as a key tool by Geroldinger [5, page 736], and then investigated by Gao [3] in a systematical way. Since then it has attracted a lot of attention in recent years (see for example [1,2,4,6,7,[9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

Minimal zero sum sequences of length four over finite cyclic groups

Plyley

Yuan

et al. 2010

Journal of Number Theory

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Text. Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n 1 g) · . . . · (n l g) where g ∈ G and n 1 , . . . ,n l ∈ [1, ord(g)], and the index ind(S) of S is defined to be the minimum of (n 1 + · · · + n l )/ord(g) over all possible g ∈ G such that g = supp(S) . The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that gcd(|G|, 6) = 1, then every minimal zero-sum sequence of length 4 has index 1.Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=BC7josX_xVs.

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“…The notion of the index of a sequence was introduced by Chapman, Freeze and Smith [1]. It was first addressed by Lemke and Kleitman (in a conjecture [10, p. 344]), used as a key tool by Geroldinger [7, p. 736], and later investigated by Gao [3] in a systematical way. Since then it has received a great deal of attention (see for example [2,5,8,9,[11][12][13][14][15][16][17][18]).…”

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confidence: 99%