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For $A\subseteq \mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ isdefined 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 $A$-weightedsum is zero. Sequences of length $E_A(n)-1$ in $\mathbb Z_n$, which do not haveany $A$-weighted zero-sum subsequence of length $n$ are called $A$-extremalsequences for the Gao constant. Such a sequence which has $n-1$ zeroes is saidto be of the standard type. When $A=U(n)$ (units in $\mathbb Z_n$) where $n$ isodd, we characterize all such sequences and show that they are of the standardtype. When $n$ is even, we give examples of such sequences which are not of thestandard type. We also characterize the $U(n)$-extremal sequences for the Gaoconstant, when $n=2^rp$, where $p$ is an odd prime.

Extremal sequences for the unit-weighted Gao constant of $\mathbb Z_n$