Polynomial Representations of Polar Codes and Decoding under Overcomplete Representations

This letter proposes an alternative expression of polar codes using polynomial representations. Polynomial representations may help to explore further algebraic properties of polar codes, same as those for convolutional codes. The relationship between message and codeword polynomials of polar codes indicates that polar codes are highly related to convolutional codes with generator polynomials 1+D and 1/(1+D). By using polynomial representations, we show the input and output bit shift properties of polar codes. This property is then employed to construct redundant trellises for overcomplete representations. Simulation results show that belief propagation (BP) decoding over the proposed overcomplete representation can achieve a significant performance gain as compared with BP decoding over the overcomplete representation using trellis permutations.