An efficient construction of polar codes based on the general partial order

Traditionally, the construction of polar codes requires intense computations to sort all bit channels. In previous works, two types of partial orders (POs) of polar codes were proposed to decrease the computations in the construction process. In this paper, a procedure is presented to employ POs with complexity O(N2), which is lower than the existing procedures, where N=2n is the code length (n≥1). To further reduce the computation complexity, in this paper, we propose a general partial order (GPO), which works at a lower dimension (nu<n) to sort bit channels. The principle of the GPO is to divide the binary expansion of bit channels into two parts: the upper part and the lower part. First, the relationships among bit channels formed from the upper part (with length nu) are completely determined. Existing algorithms are called in this new level nu when needed. Second, we prove that the ordering at this lower dimension nu can be combined with the PO ordering of bit channels formed from the lower part. Therefore, more intrinsic relationships among bit channels are obtained. Working at the lower dimension nu<n, the complexity of GPO is inherently smaller than the existing sorting algorithms. The studies show that for n=10 and the code rate R=0.5 (the worst code rate from the empirical studies), the GPO can order 82% of bit channels, which is 32% larger than that obtained from using only POs. The proposed GPO in this paper can significantly reduce computations in the construction of polar codes.