Abstract
In a series of remarkable papers ([4, 5, 26]), researchers studied the group randomness property of sequences based on binary linear codes. More precisely, they proved that the spectral distribution of random matrix form sequences based on binary linear codes is very close to the Marchenko-Pastur distribution as long as the dual distance of the codes is at least 5. The purpose of this thesis is to establish similar statements for mutually unbiased bases and more generally, for approximately mutually unbiased bases. This is the so-called group randomness of mutually unbiased bases. We will also study the group randomness property of mutually unbiased bases with respect to a different statistic arising from random matrix theory.