Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory

We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group (G,+), a set of weights W⊂Z, and an integral parameter m, the m-wise Davenport constant with weights W is the smallest integer n such that each sequence over G of length n has at least m dis...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2015-10, Vol.135, p.237-267
Main Authors: Marchan, Luz E., Ordaz, Oscar, Santos, Irene, Schmid, Wolfgang A.
Format: Article
Language:English
Subjects:
Cap set
Combinatorics
Computer Science
Davenport constant
Finite abelian group
Information Theory
Linear intersecting code
Mathematics
Number Theory
Weighted subsum
Zero-sum problem
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Summary:We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group (G,+), a set of weights W⊂Z, and an integral parameter m, the m-wise Davenport constant with weights W is the smallest integer n such that each sequence over G of length n has at least m disjoint zero-subsums with weights W. And, for an integral parameter d, the d-constrained Davenport constant with weights W is the smallest n such that each sequence over G of length n has a zero-subsum with weights W of size at most d. First, we establish a link between these two types of constants and prove several basic and general results on them. Then, for elementary p-groups, establishing a link between our constants and the parameters of linear codes as well as the cardinality of cap sets in certain projective spaces, we obtain various explicit results on the values of these constants.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2015.05.004