Periods of termwise exclusive ors of maximal length FCSR sequences

A maximal length feedback with carry shift register sequence is also called an l-sequence. Although termwise exclusive ors of l-sequences are long thought to be a type of good pseudorandom sequences, few of their statistical properties have been proved yet. This paper completely determines the period of a termwise exclusive or of several l-sequences generated by FCSRs with distinct nonprime connection integers. The main result shows that either it attains the maximum or half of it and the associated sufficient conditions are also presented. Moreover, this periodicity property also holds for generalized l-sequences of the form where ξ is a primitive root modulo odd prime number power pe and A is an integer relatively prime to p.