THE WEIGHT RECURSIONS FOR THE 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 x₁ x_qx_rx_s, then we use the notation 2-(1, q, r, s)_2n for the function. A detailed theory of equivalence of quartic MRS 2-functions in 2n variables was given in a 2020 paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called mf1 and mf2 in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions 2-(1, q, r, s)_2n (with q < r < s, say), n = s, s + 1, … can be shown to satisfy. This problem was solved for the mf1 case only in the 2020 paper. Using new ideas about “short” functions, Cusick and Cheon found formulas for the mf2 weights in a 2021 sequel to the 2020 paper. In this paper the actual recursions for the weights in the mf2 case are determined.