  We introduce the design of a set of code sequences $ \{ {\mathscr
C}_{n}^{(m)} : n\geq 1, m \geq 1 \}$, with memory order $m$ and code-length
$N=O(\phi^n)$, where $ \phi \in (1,2]$ is the largest real root of the
polynomial equation $F(m,\rho)=\rho^m-\rho^{m-1}-1$ and $\phi$ is decreasing in
$m$. $\{ {\mathscr C}_{n}^{(m)}\}$ is based on the channel polarization idea,
where $ \{ {\mathscr C}_{n}^{(1)} \}$ coincides with the polar codes presented
by Ar\i kan and can be encoded and decoded with complexity $O(N \log N)$. $ \{
{\mathscr C}_{n}^{(m)} \}$ achieves the symmetric capacity, $I(W)$, of an
arbitrary binary-input, discrete-output memoryless channel, $W$, for any fixed
$m$ and its encoding and decoding complexities decrease with growing $m$. We
obtain an achievable bound on the probability of block-decoding error, $P_e$,
of $\{ {\mathscr C}_{n}^{(m)} \}$ and showed that $P_e = O (2^{-N^\beta} )$ is
achievable for $\beta < \frac{\phi-1}{1+m(\phi-1)}$.
