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Planar functions in odd characteristic were introduced by Dembowski andOstrom in order to construct finite projective planes in 1968. They were alsoused in the constructions of DES-like iterated ciphers, error-correcting codes,and signal sets. Recently, a new notion of pseudo-planar functions in evencharacteristic was proposed by Zhou. These new pseudo-planar functions, as ananalogue of planar functions in odd characteristic, also bring about finiteprojective planes. There are three known infinite families of pseudo-planarmonomial functions constructed by Schmidt and Zhou, and Scherr and Zieve. Inthis paper, three new classes of pseudo-planar binomials are provided.Moreover, we find that each pseudo-planar function gives an association schemewhich is defined on a Galois ring.

New pseudo-planar binomials in characteristic two and related schemes