Group Actions on Binary Resilient Functions

Abstract

Let Gn,t be the subgroup of GL(n,ℤ2) that stabilizes {xℤ2n:|x|≤t}. We determine Gn,t explicitly: For 1≤t≤n−2, Gn,t=Sn when t is odd and Gn,t=〈Sn,Δ〉 when t is even, where Sn<GL(n,ℤ2) is the symmetric group of degree n and ΔGL(n,ℤ2) is a particular involution. Let ℛn,t be the set of all binary t-resilient functions defined on ℤ2n. We show that the subgroup ℤ2n⋊(Gn,t∪Gn,n−1−t)<AGL(n,ℤ2) acts on ℛn,t/ℤ2. We determine the representatives and sizes of the conjugacy classes of ℤ2n⋊Sn and ℤ2n⋊〈Sn,Δ〉. These results allow us to compute the number of orbits of ℛn,t/ℤ2 under the above group action for (n,t)=(5,1) and (6,2).