Polar Codes With Higher-Order Memory

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 $\beta < \frac{\phi-1}{1+m(\phi-1)}$.