Non-binary Polar Codes with Channel Symbol Permutations

The polar codes generated by the kernel [(1)(1) (0)(1)] are termed as unitary polar codes. It has been shown that q-ary unitary polar codes polarize arbitrary q-ary input channels if q is a prime number. However, this is in general not true if q is not a prime number. To achieve polarization for arbitrary q-ary input channels, the conventional approaches were to modify the kernel using permutations of alphabet inside the kernel, or assign non-unitary entries in the kernel if q is a prime power. In this paper, we propose a different approach by using the concept of channel symbol permutations. We show that unitary polar codes can still polarize arbitrary q-ary input channels if the code symbols from the unitary polar code are randomly mapped to the channel symbols before transmission. By using the channel randomization concept, we show that unitary polar codes polarize arbitrary q-ary input randomized channels. If q is a prime power, some sequences of fixed channel symbol permutations are given that, with unitary polar codes, guarantee polarization for arbitrary q-ary input channels.