  The linear complexity of a sequence $s$ is one of the measures of its
predictability. It represents the smallest degree of a linear recursion which
the sequence satisfies. There are several algorithms to find the linear
complexity of a periodic sequence $s$ of length $N$ (where $N$ is of some given
form) over a finite field $F_q$ in $O(N)$ symbol field operations. The first
such algorithm is The Games-Chan Algorithm which considers binary sequences of
period $2^n$, and is known for its extreme simplicity. We generalize this
algorithm and apply it efficiently for several families of binary sequences.
Our algorithm is very simple, it requires $\beta N$ bit operations for a small
constant $\beta$, where $N$ is the period of the sequence. We make an analysis
on the number of bit operations required by the algorithm and compare it with
previous algorithms. In the process, the algorithm also finds the recursion for
the shortest linear feedback shift-register which generates the sequence. Some
other interesting properties related to shift-register sequences, which might
not be too surprising but generally unnoted, are also consequences of our
exposition.
