On a Family of Planar Mappings

A mapping j : GF(q) --> GF(q) is called planar if for every nonzero a. E GF(q) the difference mapping D I.a : x I-t j(x + a) - j(x) - j(a) is a permutation of GF(q). In this note we show that for certain choices of (3, 'Y the mapping j(:z;) = Tl'({3x CJ + 1 ) +-fX~ is planar on GF(q~-J).