An application of coding theory to estimating Davenport constants

Abstract

Abstract. We investigate a certain well-established generalization of the Davenport constant. For j a positive integer (the case j = 1, is the classical one) and a finite Abelian group (G, +, 0), the invariant D j (G) is defined as the smallest ℓ such that each sequence over G of length at least ℓ has j disjoint non-empty zero-sum subsequences. We investigate these quantities for elementary 2-groups of large rank (relative to j). Using tools from coding theory, we give fairly precise estimates for these quantities. We use our results to give improved bounds for the classical Davenport constant of certain groups.

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

“…We use some ad-hoc terminology based on the one introduced in [28]. Thus, (asymptotically) p-upper-bounding functions are the functions in the (asymptotic) upper bounds for the rate k/n of a p-linear code as a function of its normalized minimal distance d/n.…”

“…In this section, we develop the link between fully-weighted zero-sum problems and problems on linear codes that was mentioned already in the introduction. We recall that for elementary 2-groups, this link was already known; note that in fact for these groups the fully-weighted problem coincides with the classical one, and the connection came up in that context (see [9,28]). Furthermore, we recall that for elementary 3-groups the fully-weighted problem coincides with the plus-minus weighted problem, which is of particular interest.…”

On the Erdős–Ginzburg–Ziv constant of finite abelian groups of high rank

“…More information can be found in the surveys [9,13]. Both the Davenport constant and the Erdős-Ginzburg-Ziv constant have found far reaching generalizations, and for these generalized versions, the precise values have been determined for groups with rank at most two (see [15,Section 6.1], [7], [14,Theorem 5.2], [17]). …”

“…Then (s(G) − 1)/2 equals the maximal size of a cap in the affine space F r 3 . The maximal size of such caps has been studied in finite geometry for decades, and the precise value is known so far only for r 6 (see [18,2]). The connection to affine caps will be addressed in greater detail in Section 4.…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…As another example, Nathanson [239] employed König's infinity lemma on the existence of infinite paths in certain infinite graphs, and introduced a new class of additive bases, and also a generalization of the Erdős-Turán conjecture about additive bases of the positive integers. In [245] the authors employed tools from coding theory to estimating Davenport constants. Also, in [15,230,231,268,319] informationtheoretic techniques are used to study sumset inequalities.…”