The Linear Complexity of Whiteman's Generalized Cyclotomic Sequences of Period $p^{m+1}q^{n+1}$

摘要

In this paper, we mainly get three results. First, let $p$, $q$ be distinct primes with $gcd ((p-1)p,(q-1)q)=gcd (p-1,q-1)=e$ ; we give a method to compute the linear complexity of Whiteman's generalized cyclotomic sequences of period $p^{m+1}q^{n+1}$. Second, if $e=4$, we compute the exact linear complexity of Whiteman's generalized cyclotomic sequences. Third, if $p equiv q equiv 5~({rm mod}~8)$, $gcd (p-1, q-1)=4$, and we fix a common primitive root $g$ of both $p$ and $q$, then $2in H_{0}=(g)$, which is a subgroup of the multiplicative group $Z_{pq}^{ast}$, if and only if Whiteman's generalized cyclotomic numbers of order 4 depend on the decomposition $pq=a^{2}+4b^{2}$ with $4vert b$.