The odd-cycle naive-dimension conjecture

Fix an odd cycle $C_m$, with $m=2d+1$, and let $k_{\min}(C_m)$ denote the minimum dimension of an isometric embedding of $C_m$ into an abelian Cayley graph. Encode the edge-generator dependencies by the binary code

$$D=\Bigl\{x\in\mathbb{F}_2^m:\ \sum_{i:x_i=1}s_i=0\Bigr\}.$$

The code contains the all-ones vector, and for every cyclic interval $W\subseteq\mathbb{Z}_m$ isometry requires

$$\min_{x\in D}|W\triangle x|=\min\bigl(|W|,m-|W|\bigr).$$

Odd-cycle naive-dimension conjecture. The cyclic interval lemma holds for every odd $m$; consequently,

$$k_{\min}(C_m)=m-1$$

for every odd cycle. The claim is proved in the paper for every odd $m\leq17$ and for any further odd $m$ for which the cyclic interval lemma holds; the general case remains open.