Source and Channel Polarization over Finite Fields and Reed-Solomon Matrices

Polarization phenomenon over any finite field Fq with size q being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arıkan for the binary field, as well as its extension to a prime field by Şaşoğlu, Telatar, and Arıkan. In this paper, a necessary and sufficient condition of a matrix over a finite field Fq is shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arıkan and Telatar is generalized to arbitrary finite field. It is also shown that the asymptotic error probability of polar codes is improved by using the Reed-Solomon matrices, which can be regarded as a natural generalization of the 2 × 2 binary matrix used in the original proposal by Arıkan.