Design of Short Polar Codes for SCL Decoding

The problem of designing polar-like codes for successive cancellation list (SCL) decoding algorithm is considered. A novel code design algorithm aimed to minimize both successive cancellation (SC) and maximum-likelihood (ML) decoding error probabilities is introduced. The algorithm performs optimization of a precoding matrix for the polarization transformation matrix to guarantee preselected minimum distance and the lowest possible SC decoding error probability to the resulting code. Constructed precoded polar codes are decoded as polar codes with dynamic frozen bits. Numerical results show that the proposed codes of lengths 32 and 64 significantly reduce the frame error rate (FER) compared to the original polar codes under the SCL decoding starting with the list size 4. The gain increases with the list size. The proposed codes have a much lower SC decoding error probability than extended Bose-Chaudhuri-Hocquenghem (e-BCH) codes and their polar subcodes without sacrificing the ML performance. In case of the code length 64, the constructed precoded polar codes demonstrate a FER reduction compared to the polar subcodes of e-BCH codes under the SCL decoding with the list size up to 16 and demonstrate a comparable FER starting with the list size 32.