  Sequences with optimal correlation properties are much sought after for
applications in communication systems. In 1980, Alltop (IEEE Trans. Inf. Theory
26(3):350-354, 1980) described a set of sequences based on a cubic function and
showed that these functions were optimal with respect to known bounds on auto
and crosscorrelation. Subsequently these sequences were used to construct
mutually unbiased bases, a structure of importance in quantum information
theory. The key feature of this cubic function is that its difference function
is a planar function. Functions with planar difference functions have been
called \emph{Alltop functions}. This paper provides a new family of Alltop
functions and establishes the use of Alltop functions for construction of
sequence sets and MUBs.
