(2ⁿ,2ⁿ,2ⁿ,1)-relative difference sets and their representations

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We show that every (2ⁿ,2ⁿ,2ⁿ,1)-relative difference set D in ₄ⁿ relative to ₂ⁿ can be represented by a polynomial f(x)∈ ₂ₙ[x], where f(x+a)+f(x)+xa is a permutation for each nonzero a. We call such an f a planar function on ₂ₙ. The projective plane Π obtained from D in the way of Ganley and Spence ganleyᵣelative₁975 is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on ₂ₙ 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∈₂ₙ.