Research On Polar Codes And Polar Kernels

Up to now,polar codes are the only coding scheme which can achieve Shannon Limit by using theoretical proofs.Based on the channel polarization phenomenon,through channel combining and channel splitting,original channels which are independent and have the same error probabilities become virtual channels which are interdependent and have different error probabilities.Finally,each channel becomes either a noiseless chan-nel or a completely noisy channel.Polarization phenomenon is induced by polar matrices,but it is difficult to directly construct matrices of large dimensions.Therefore,it is worth-while to study how to synthesize large matrices by using some small polar matrices with good performance.Firstly,this paper proposes a fast method to calculate Bhattacharyya parameters.These parameters can be seen as the classification indicator of small polar matrices.These calculation formulas have similar structures,so they can reduce computational complex-ities and they are very suitable to calculate final results by using mathematical software.Matrices will be grouped into the same classification if they have the same Bhattacharyya parameters.Secondly,this paper compares different polar matrices based on classification re-sults.Based on the discrete degree of Bhattacharyya parameters,this paper proposes a new parameter,named as comparison parameter,to measure the performance of polar ma-trices.The larger value of comparison parameter indicates the better performance of the polar matrix.As a result,the most suitable classification can be selected under different conditions by comparing comparison parameters.Finally,this paper constructs large polar matrices by using the Kronecker products of selected small polar matrices.This paper proposes two methods to construct large po-lar matrices,namely,dynamic multi-kernels and hybrid multi-kernels.In each stage of a Kronecker product,factor polar kernels are dynamically selected to make the comparison parameter of this stage reach the maximum.In most cases,dynamic multi-kernels can improve performance.Different from dynamic multi-kernels,hybrid multi-kernels do not strictly follow the rule of Kronecker product operation and can be seen as results of a Kronecker-like product operation.In each stage,different polar matrices will be selected for different virtual channels based on channel information.As a result,constituent ma-trices in the same stage may be different.Compared with dynamic multi-kernels,hybrid multi-kernels can further improve performance in terms of error exponents,comparison parameters and block error rate upper bounds.