The weight recursions for the 2-rotation symmetric quartic Boolean functions

Abstract

A Boolean function in $ n $ variables is 2-rotation symmetric if it is invariant under even powers of $ \rho(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1) $, 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 $ \rho^2 $ to a single monomial. If the quartic MRS 2-function in $ 2n $ variables has a monomial $ x_1 x_q x_r x_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, \ldots $ 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.