Polar Codes Bhattacharyya Parameter Generalization

Recently, polar codes were proposed by Arikan to achieve optimum channel capacity given by Shannon theorem with low encoding and decoding complexity. Polar code construction depends on two main foundation criteria which are kernel matrix and Bhattacharyya parameter. They are related to each other, therefore the selection method for both affects the performance of polar code. Firstly, in this study, the derivations of the bounds for Bhattacharyya parameter are proved and generalised together with a proposed method to select the best kernel matrix to achieve the optimum capacity. Then, recursive channel transformations and successive cancellation decoding of the selected 3 × 3 best kernel matrix are proved. Furthermore, a general formula for polar code complexity and hardware implementation has been discussed. Simulation results show that the achievable bit error rate for the proposed methodology of selection is the same as some existing methods with the same order of complexity, which indicates its effectiveness. Polar code performance for the selected 3 × 3 best kernel matrix is improved as the code length increases. Moreover, this proposed method is general to be for higher dimension kernel matrices.