On the connection between mutually unbiased bases and orthogonal Latin squares

@article{Paterek2010OnTC,
title={On the connection between mutually unbiased bases and orthogonal Latin squares},
author={T. Paterek and M. Pawlowski and M. Grassl and {\vC}. Brukner},
journal={Physica Scripta},
year={2010},
volume={2010},
pages={014031}
}

We offer a piece of evidence that the problems of finding the number of mutually unbiased bases (MUB) and mutually orthogonal Latin squares (MOLS) might not be equivalent. We study a particular procedure that has been shown to relate the two problems and generates complete sets of MUB in power-of-prime dimensions and three MUB in dimension six. For these cases, every square from an augmented set of MOLS has a corresponding MUB. We show that this no longer holds for certain composite dimensions.