The throughput of successive cancellation decoding of polar codes can be improved through simplified decoders that identify specific constituent codes in the decoding tree. The identified codes include rate-0, rate-1, repetition, and single parity-check codes. In this work, constituent codes that belong to the family of first-order Reed-Muller codes, and their sub-codes, are also identified in the decoding tree. Alternative decoding schemes that utilize the structure of Reed-Muller codes are incorporated into successive cancellation decoding. Simulation results show that such an approach can improve both the block error rate performance as well as the decoding latency of polar codes.

A practical rate-matching system for constructing rate-compatible polar codes is proposed. The proposed polar code circular buffer rate-matching is suitable for transmissions on communication channels that support hybrid automatic repeat request communications, as well as for flexible resource-element rate-matching on single transmission channels. Our proposed circular buffer rate matching scheme also incorporates a bit-mapping scheme for transmission on bit-interleaved coded modulation (BICM) channels using higher order modulations. An interleaver is derived from a puncturing order obtained with a low complexity progressive puncturing search algorithm on a base code of short length, and has the flexibility to achieve any desired rate at the desired code length, through puncturing or repetition. The rate-matching scheme is implied by a two-stage polarization, for transmission at any desired code length, code rate, and modulation order, and is shown to achieve the symmetric capacity of BICM channels. Numerical results on AWGN and fast fading channels show that the rate-matched polar codes have a competitive performance when compared with the spatially-coupled quasi-cyclic LDPC codes or LTE turbo codes, while having similar rate-dematching storage and computational complexities.

A design of rate-compatible polar codes suitable for HARQ communications is proposed in this paper. An important feature of the proposed design is that the puncturing order is chosen with low complexity on a base code of short length, which is then further polarized to the desired length. A practical rate-matching system that has the flexibility to choose any desired rate through puncturing or repetition while preserving the polarization is suggested. The proposed rate-matching system is combined with channel interleaving and a bit-mapping procedure that preserves the polarization of the rate-compatible polar code family over bit-interleaved coded modulation systems. Simulation results on AWGN and fast fading channels with different modulation orders show the robustness of the proposed rate-compatible polar code in both Chase combining and incremental redundancy HARQ communications.

Polar codes proposed by Arikan are based on a linear kernel of dimension two with exponent 0.5. In this paper, binary kernels of maximum exponents of dimensions up to 16 are presented except for the case of dimension 12 where the maximum exponent is shown to be attained by either a constructed linear kernel or a possible nonlinear kernel with a specified partial distance sequence. The results show that the minimum dimension for which there exists a kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arikan, is 14. For dimensions 14, 15, 16, discussed by Presman et al., along with 13, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.

Polar codes are constructed based on kernels with polarizing properties. The performance of a polar code is characterized asymptotically in terms of the exponent of its kernel. The pioneering work of Arıkan on polar codes is based on a linear kernel of dimension two and exponent 0.5. In this paper, constructions of linear and nonlinear binary kernels of dimensions up to 16 are presented. The kernels are obtained using computer search or by shortening longer kernels obtained by computer search and a computer program is used to determine their exponents. It is proved that the constructed kernels have maximum exponents except in the case of nonlinear kernels of dimension 12 where it is demonstrated that the maximum exponent either equals that of the presented construction or assumes another specified value. The results show that the minimum dimension for which there exists a linear kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arıkan, is 15, while this minimum dimension is 14 for nonlinear kernels. Furthermore, it is shown that there is a linear kernel with maximum exponent up to dimension 11. For dimensions 13, 14, 15, and 16, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.