Extremal sequences for the unit-weighted Gao constant of ${\mathbb{Z}}_n$

Abstract

For $A\subseteq {\mathbb{Z}}_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in ${\mathbb{Z}}_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero. Sequences of length $E_A(n)-1$ in ${\mathbb{Z}}_n$, which do not have any $A$-weighted zero-sum subsequence of length $n$ are called $A$-extremal sequences for the Gao constant. Such a sequence which has $n-1$ zeroes is said to be of the standard type. When $A=U(n)$ (units in ${\mathbb{Z}}_n$) where $n$ is odd, we characterize all such sequences and show that they are of the standard type. When $n$ is even, we give examples of such sequences which are not of the standard type. We also characterize the $U(n)$-extremal sequences for the Gao constant, when $n=2^{r}p$, where $p$ is an odd prime.

14 pages. arXiv admin note: text overlap with arXiv:2203.02665