  Planar functions over finite fields give rise to finite projective planes.
They were also used in the constructions of DES-like iterated ciphers,
error-correcting codes, and codebooks. They were originally defined only in
finite fields with odd characteristic, but recently Zhou introduced
pesudo-planar functions in even characteristic which yields similar
applications. All known pesudo-planar functions are quadratic and hence they
give presemifields. In this paper, a new approach to constructing quadratic
pseudo-planar functions is given. Then five explicit families of pseudo-planar
functions are constructed, one of which is a binomial, two of which are
trinomials, and the other two are quadrinomials. All known pesudo-planar
functions are revisited, some of which are generalized. These functions not
only lead to projective planes, relative difference sets and presemifields, but
also give optimal codebooks meeting the Levenstein bound, complete sets of
mutually unbiased bases (MUB) and compressed sensing matrices with low
coherence.
