Extended quadratic truncated rotation symmetric Boolean functions

Abstract

A Boolean function in n variables $x_1,\dots ,x_n$ is rotation symmetric (RS) if the function is invariant under cyclic rotation of the variables. If the function is generated by a single monomial it is called monomial rotation symmetric (MRS). An MRS function is called truncated rotation symmetric (TRS) if the expansion for the n terms of the MRS function is stopped at the first term where $x_n$ occurs. This paper studies extended TRS functions, which are generated by adding the next monomial in the expansion of the MRS function to the TRS function. For example, the MRS function in 5 variables generated by $x_1{x}_3$ gives the TRS function $x_1{x}_3+x_2{x}_4+x_3{x}_5$ and the extended TRS function $x_1{x}_3+x_2{x}_4+x_3{x}_5+x_1{x}_4$. It is shown that the Hamming weights of any quadratic TRS function satisfy the same linear recursion as the weights of the corresponding extended TRS function, and also that the weights for the two functions are very frequently equal. The problem of finding the Dickson form (very difficult for a general quadratic function) for any quadratic extended TRS function is solved and an explicit generating function for the weights of any quadratic extended TRS function is found.