Refining the search space of Alltop monomials

Alltop functions have applications to code-division multiple access (CDMA) systems and mutually unbiased bases (MUBs). Alltop functions construct MUBs and CDMA signal sets, and this motivates the continued search for further Alltop functions. The discovery of further Alltop functions is hindered by the computational complexity of verification. This paper narrows the search space through the introduction of a characteristic cubic polynomial and provides a verification process for Alltop monomials with a reduced computational complexity. Computational results lead to the conjecture that there are no further Alltop monomials.