Zero-sum problems in finite abelian groups and affine caps

coauthors C. Elsholtz, A. Geroldinger, S. Kubertin, L. Rackham.

Quarterly Journal of Mathematics, 58 (2007), 159-186.



For a finite abelian group G let s(G) denote the smallest integer l such that every sequence S over G of length |S| >= 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 = Cnr but they respect the structure of the group. In particular, we show s(Cn4) >= 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.

Download the preprint as pdf.

Update 1:

There are some improvements over the lower bounds given in Lemma 5.3. There is a:

496-cap in AG(8,3)
2432-cap in AG(10,3)
5488-cap in AG(11,3)
12928-cap in AG(12,3)

These affine caps can be obtained by Mukhopadhyay's doubling construction applied on the projective caps in the next smaller dimension. This construction is e.g. mentioned as Theorem 4 in Recursive constructions for large caps.

See my Table with lower bounds for projective caps. Detailed informations as well as upper bounds can also be found at the MinT data base of linear codes, orthogonal arrays, OOA and tms-nets, in the OA-tables, for linear OA of base 3, and strength 3 (= ternary projective caps).

Update 2:

The maximal size of cap in AG(6,3) is meanwhile kown to be 112, see:
Aaron Potechin, Maximal caps in AG(6,3), Designs, Codes and Cryptography 46 (2008), 243-259

| home | List of publications |