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The Walsh transform  QˆQ^  of a quadratic function  Q:Fpn→FpQ:Fpn→Fp  satisfies  |Qˆ(b)|∈{0,pn+s2}|Q^(b)|∈{0,pn+s2}  for all  b∈Fpnb∈Fpn , where  0≤s≤n−10≤s≤n−1  is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class  C1C1  is defined for arbitrary n as  C1={Q(x)=Trn(∑⌊(n−1)/2⌋i=1aix2i+1):ai∈F2}C1={Q(x)=Trn(∑i=1⌊(n−1)/2⌋aix2i+1):ai∈F2} , and the larger class  C2C2  is defined for even n as  C2={Q(x)=Trn(∑(n/2)−1i=1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2}C2={Q(x)=Trn(∑i=1(n/2)−1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2} . For an odd prime p, the subclass  DD  of all p-ary quadratic functions is defined as  D={Q(x)=Trn(∑⌊n/2⌋i=0aixpi+1):ai∈Fp}D={Q(x)=Trn(∑i=0⌊n/2⌋aixpi+1):ai∈Fp} . We determine the generating function for the distribution of the parameter s for  C1,C2C1,C2  and  DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case  p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to  C1C1 and  C2C2  in terms of a generating function

Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension