Polar Codes With Higher-Order Memory

Abstract: We introduce the design of a set of code sequences $${ {\mathscr C}<em>{n}^{(m)} : n\geq 1, m \geq 1 }$$, with memory order $m$ and code-length $N=O(\phi<sup>n)$, where $\phi \in (1,2]$ is the largest real root of the polynomial equation $F(m,\rho)=\rho<sup>m-\rho<sup>{m-1}-1$ and $\phi$ is decreasing in $m$. ${ {\mathscr C}</em>{n}<sup>{(m)}}$ is based on the channel polarization idea, where ${ {\mathscr C}<em>{n}^{(1)} }$ coincides with the polar codes presented by Ar\i kan and can be encoded and decoded with complexity $O(N \log N)$. ${ {\mathscr C}</em>{n}^{(m)} }$ 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 $\beta &lt; \frac{\phi-1}{1+m(\phi-1)}$.