Research On General Construction Algorithm Of Polar Codes

The core of the construction of the polar code is the polarization of the channel and it is the only code that can theoretically prove that the Shannon limit can be achieved.It combines channel splitting and channel merging,so that the quality of all bit channels tends to two extremes,the capacity of some channels tends to zero,that is,it tends to a purely noisy channel,while the capacity of the other part of channels tends to 1,that is,tends to a noiseless channel,thereby achieving polarization of the channel.The code length of the polar codes is usually the power of two.In some practical applications,the code length and code rate of the polar codes are arbitrary,that is,the original polar codes's length is limited.Therefore,the original polar codes can be punctured to achieve the actual requirements.Since the quality of the underlying channel of the polar codes changes after puncturing,that is,the quality of each underlying channel is no longer identical.Therefore,the set of information bits selected before puncturing is not necessarily suitable for the polar codes after puncturing,so it is necessary to reconstruct the polar codes after puncturing.The existing construction algorithm only GA can reconstruct the punctured polar codes,but the underlying channel of the GA structure must be the Gaussian channel(AWGN).Therefore,for other types of underlying channels,it is currently impossible to reconstruct them.For example,the binary symmetric channel(BSC),the fading channel,and the like.This paper provides two improved construction algorithms,namely:the improved BEC construction algorithm and the improved Tal-Vardy construction algorithm.The improved BEC construction algorithm can not only reconstruct the punctured polar code of the BEC channel of the underlying channel,but also construct the polar code of the BEC channel with independent underlying channels but different quality,thereby solving the problem of reconstructing the punctured polar code of the BEC channel.The improved Tal-Vardy construction algorithm can not only reconstruct the punctured polar codes of the Binary-input memoryless symmetric(BMS)channel of the underlying channel,but also construct the polar code of the BMS channels with independent underlying channel but different distribution,thus solving the problem of reconstructing the punctured polar code of the BMS channel.Finally,simulation results show that the FER performance of the reconstructed punctured polar code using the improved BEC construction algorithm or the improved Tal-Vardy construction algorithm is significantly better than the FER performance of the constructed punctured polar code using the original BEC construction algorithm or the original Tal-Vardy construction algorithm.Therefore,the improved Tal-Vardy algorithm is a general construction algorithm of the polar codes,that is,the polar code of the BMS channel of any independent and identical distribution can not only be constructed for the underlying channel,but also polar codes of BMS channels that are independent but of different distribution are constructed for the underlying channel.