Fundamental study on theory and applications of channel polarization

极地码与相继消除译码相结合，作为纠错码引起了研究人员的关注，已被证明可以在长码长的限制下饱和任何无记忆信道的对称容量，同时编码和译码需要相对较低的计算复杂度。在本研究项目中，为了获得有助于构造性能更好的码的认识，我们阐明了信道极化现象，为极性码的高性能奠定了数学基础。通过改进已有的关于译码错误概率的渐近结果，我们得到了一个码率相关形式的渐近译码错误概率结果。我们还研究了非二进制字母表上的信道极化，得到了一个可逆的充要条件。FQ上的矩阵以诱导通道极化。

Polar codes combined with successive-cancellation decoding, that were newly proposed in 2008, have been attracting attentions of researchers as error-correcting codes that have been proven to saturate the symmetric capacity for any memoryless channel in the limit of long codelength, while requiring relatively low computational complexity in encoding and decoding. In this research project, aiming at obtaining understandings that will be useful in constructing codes with better performance, we have elucidated the phenomenon of channel polarization, which lays the mathematical basis for high performance of polar codes. We have obtained a result on asymptotic decoding error probability in a coderate-dependent form, by refining existing asymptotic results on decoding error probability. We have also studied channel polarization over non-binary alphabet, and have obtained a necessary and sufficient condition for an invertible ?×? matrix on Fq to induce channel polarization.