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Zero-sum problems in finite abelian groups and affine caps. (English) Zbl 1205.11028

Summary: For a finite abelian group \(G\) let \(s(G)\) denote the smallest integer \(l\) such that every sequence \(S\) over \(G\) of length \(| S|\geq l\) has a zero-sum subsequence of length \(\exp(G)\). We derive new upper and lower bounds for \(s(G)\) and all our bounds are sharp for special types of groups. The results are not restricted to groups \(G\) of the form \(G = C^r_n\) but they respect the structure of the group. In particular, we show \(s(C^4_n) \geq 20n-19\) for all odd \(n\) which is sharp if \(n\) is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.

MSC:

11B83 Special sequences and polynomials
20K01 Finite abelian groups
51E22 Linear codes and caps in Galois spaces

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