( 2 n , 2 n , 2 n , 1 ) ‐Relative Difference Sets and Their Representations

We show that every ( 2 n , 2 n , 2 n , 1 ) ‐relative difference set D in Z 4 n relative to Z 2 n can be represented by a polynomial f ( x ) ∈ F 2 n[ x ] , where f ( x + a ) + f ( x ) + x a is a permutation for each nonzero a . We call such an f a planar function on F 2 n . The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on F 2 nwith exactly two elements in its image set and f ( 0 ) = 0 is planar, if and only if, f ( x + y ) = f ( x ) + f ( y ) for any x , y ∈ F 2 n.