(2n, 2n, 2n, 1)‐Relative Difference Sets and Their Representations. Issue 12 (16th April 2013) - Zhou, Yue
Abstract We show that every (2n, 2n, 2n, 1)‐relative difference set D in Z4n relative to Z2n can be represented by a polynomial f(x)∈F2n[x], where f(x+a)+f(x)+xa is a permutation for each nonzero a. We call such an f a planar function on F2n. 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 F2n with 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∈F2n.