Fast multi-dimensional polar encoding and decoding

New decoding methods for polar codes are derived from a multi-dimensional realization of polar codes. The notion of polar transformation is extended into higher dimensional polar transformations. It is shown that two dimensional polar transformation leads to a rate-adaptive doubly concatenated polar code. The results are generalized to n-dimensional construction and decoding of polar codes, where N = 2n is the length of the code. The proposed structure for successive cancellation decoding, based on the n-dimensional construction of polar codes, results in efficient decoding latency and space complexity. Soft successive decoding of polar codes and iterative detection and decoding of compound polar codes are implemented using the proposed structure for multi-dimensional decoding of polar codes.