Equivalence of 2-rotation symmetric quartic Boolean functions

A Boolean function in n variables is 2-rotation symmetric if it is invariant under even powers of ρ(x₁,……,x_n)=(x₂,…,x_n,x₁), but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ² to a single monomial. If the quartic MRS 2-function in 2n variables has a monomial x₁x_qx_rx_s, then we use the notation 2-(1, q, r, s)_2n for the function. This paper gives a detailed theory of equivalence of quartic MRS 2-functions in 2n variables. Such a theory was provided for the cubic MRS 2-functions in two 2015 papers of Cusick and Johns. As in the earlier papers, the two main topics in the theory are describing the affine equivalence classes of the functions under certain groups of permutations; and giving details of the linear recursions that the Hamming weights of any sequence of functions 2-(1, q, r, s)_2n (with q < r < s, say), n=s,s+1,… can be shown to satisfy. The discussion for both of these topics uses new ideas because the quartic theory naturally divides into two cases.