Construction of orthogonal extraordinary supersquares and mutually unbiased bases

This article studies the connection between mutually orthogonal extraordinary supersquares and mutually unbiased bases. Firstly, we generalize extraordinary supersquares defined by Ghiu et al. to arbitrary order $d$.For order $d=p^a$ being a single prime power, we construct $d+1$ mutually orthogonal extraordinary supersquares; for arbitrary order $d$, we construct $p_1^{a_1}+1$ mutually orthogonal extraordinary supersquares, where $p_1^{a_1}$ is the minimal prime power dividing $d$.In addition, this article sets up the connection between mutually orthogonal extraordinary supersquares and mutually unbiased bases for arbitrary $d$. Then we provide a new proof for the known lower bound of mutually unbiased bases.