Abstract
Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.
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