On generalized Erd\H{o}s-Ginzburg-Ziv constants for $\mathbb{Z}_2^d$

TL;DR

This paper determines exact values of the generalized Erd ext{"o}s-Ginzburg-Ziv constants for certain binary vector spaces and explores their connections to binary linear codes and forbidden weight codes.

Contribution

It provides exact calculations of $s_{2m}( ext{Z}_2^d)$ for specific dimensions and lengths, advancing understanding of zero-sum problems in finite abelian groups.

Findings

Exact values of $s_{2m}( ext{Z}_2^d)$ for $d \,\leq\, 2m+1$

Connections established between zero-sum constants and binary linear codes

Insights into codes without forbidden weights

Abstract

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erd\H{o}s-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We find exact values of $s_{2m}(\mathbb{Z}_2^d)$ for $d \leq 2m+1$. Connections to linear binary codes of maximal length and codes without a forbidden weight are discussed.