Existence of self-reverse-dual \(m\)-sequences (Q1096586)

Let \(f\) be an \(n\)-ary function over \(\text{GF}(2)\). If the \(n\)-stage shift register sequence with feedback logic \(f\) has period \(2^ n\) then \(f\) is said to be an \(n\)-stage \(M\)-logic. If \(f(x_ 1,\dots,x_ n)=f(x_ 1,x_ n+1,\dots,x_ 2+1)\) for any \(x_ 1,\dots,x_ n\) in \(\text{GF}(2)\), then \(f\) is said to be a self-reverse-dual function (SRD function). It is well-known that \(n\)-stage SRD \(M\)-logics do not exist for even \(n\). This paper deals with the case of odd \(n\) and proves the following: There exists an \(n\)-stage SRD \(M\)-logic for every odd \(n>2\); and the number of \(n\)-stage SRD \(M\)-logics is a multiple of \(2^{(n+1)/2}\) for every odd \(n>2\).