On the Davenport constant and on the structure of extremal zero-sum free sequences

Let $G = C_{n_1} \oplus ... \oplus C_{n_r}$ with $1 < n_1 \t ... \t n_r$ be a finite abelian group, $\mathsf d^* (G) = n_1 + ... + n_r - r$, and let $\mathsf d (G)$ denote the maximal length of a zero-sum free sequence over $G$. Then $\mathsf d (G) \ge \mathsf d^* (G)$, and the standing conjecture is that equality holds for $G = C_n^r$. We show that equality does not hold for $C_2 \oplus C_{2n}^r$, where $n \ge 3$ is odd and $r \ge 4$. This gives new information on the structure of extremal zero-sum free sequences over $C_{2n}^r$.