  At the prime $2$, Behrens, Hill, Hopkins and Mahowald showed $M_2(1,4)$
admits $32$-periodic $v_2$-self-map and more recently authors joint with
Mahowald showed $A_1$ also admits $32$-periodic $v_2$-self-map. This leads to
the question, whether there exists a finite $2$-local complex with periodicity
less than $32$. This paper will answer the question by producing a class of
finite $2$-local spectra $\widetilde{\mathcal{Z}}$ all of which admit
$1$-periodic $v_2$-self-map.

Algebraic Topology