(2n, 2n, 2n, 1)‐Relative Difference Sets and Their Representations. Issue 12 (16th April 2013) - Zhou, Yue
Abstract We show that every <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szw3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />(2n, 2n, 2n, 1)</alternatives>‐relative difference set D in <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szxn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Z4n</alternatives> relative to <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szsf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Z2n</alternatives> can be represented by a polynomial <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25szt0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />f(x)∈F2n[x]</alternatives>, where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t03d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />f(x+a)+f(x)+xa</alternatives> is a permutation for each nonzero a. We call such an f a planar function on <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t062" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />F2n</alternatives>. 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 <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t00r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />F2n</alternatives> with exactly two elements in its image set and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t3fg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />f(0)=0</alternatives> is planar, if and only if, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t3dx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />f(x+y)=f(x)+f(y)</alternatives> for any <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3j25t3hk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />x, y∈F2n</alternatives>.