Properties C and D fail for $C_n^r$ whenever $n \equiv 2 \bmod 4, n\geq 6$ and $r\geq 4$

Abstract

Let $s(C_n^r)$ be the Erdős--Ginzburg--Ziv constant (zero-sums of length exactly $n$) and $η(C_n^r)$ the short zero-sum constant. Properties C and D of Gao et al.\ assert that every sequence of the extremal length without the relevant zero-sums has the form $T^{n-1}$. For even $n$ they imply $s(C_n^r)=2^r(n-1)+1$ and $η(C_n^r)=(2^r-1)(n-1)+1$. G.~Wang recently disproved both for $C_6^r$, $r\ge4$. We prove that both fail for $C_n^r$ for every $n\equiv2\pmod4$ with $n\ge6$ and every $r\ge4$. More precisely, for these $n$ we have: $s(C_n^r)\ge2^r(n-1)+2^{r-2}+1$.