In recent years, constraint solvers show increasing use in solving various open combinatorial problems, e.g., from Ramsey theory or synthesis of combinatorial designs. The similar approach can be applied to some problems related to binary linear codes, which form one of the largest families of error correcting codes used both in coding theory and in various practical applications. Thanks to a simple algebraic structure of such codes it is possible to study them using a wide range of methods. Note that even codes with the same basic parameters (length n, dimension k, minimum code distance d) can show different error correction performance, i.e., the ability to correct errors which appear in a noisy channel. In the paper, we formulate the problem of finding binary linear codes with good error correction performance as a constraint optimization problem and explore the effectiveness of modern constraint solvers on it, including SAT, MaxSAT, and CP solvers. Using the respective solvers and parallel computing, for several values of n, k, d we found the codes which are significantly better than the known in terms of their practical performance.

A novel search method for large polarization kernels is proposed. The algorithm produces a kernel with given partial distances by employing the depth-first search combined with the computation of coset leaders weight tables and sufficient conditions of code non-equivalence. Using the proposed method, we improved all existing lower bounds on the maximum error exponent for kernels of size from 17 to 29. We also obtained kernels which admit low complexity processing by the recently proposed recursive trellis algorithm. Numerical results demonstrate the advantage of polar codes with the obtained kernels compared with shortened polar codes and polar codes with small kernels.

A new search method for large polarization kernels is proposed. The algorithm produces a kernel with given partial distances by employing depth-first search combined with some methods for search space reduction. Using the proposed method, we improved almost all existing lower bounds on the maximum rate of polarization for kernels of size from 17 to 27. We also obtained kernels which admit low complexity processing by the recently proposed recursive trellis algorithm. Numerical results demonstrate the advantage of polar codes with the proposed kernels compared with shortened polar codes and polar codes with small kernels.

A decoding algorithm for polar (sub)codes with binary 2t×2tpolarization kernels is presented. It is based on the window processing (WP) method, which exploits the linear relationship of the polarization kernels and the Arikan matrix. This relationship enables one to compute the kernel input symbols probabilities by computing the probabilities of several paths in Arikan successive cancellation (SC) decoder. In this paper we propose an improved version of WP, which has significantly lower arithmetic complexity and operates in log-likelihood ratios (LLRs) domain. The algorithm identifies and reuses common subexpressions arising in computation of Arikan SC path scores. The proposed algorithm is applied to kernels of size 16 and 32 with improved polarization properties. It enables polar (sub)codes with the considered kernels to simultaneously provide better performance and lower decoding complexity compared with polar (sub)codes with Arikan kernel.

We propose a reduced complexity algorithm for computing log-likelihood ratios (LLRs) needed for successive cancellation (SC) decoding of polar codes with 2t× 2tpolarization kernels. This algorithm is applied to some polarization kernels of length 16 and 32 with high polarization rate. The complexity reduction is achieved by exploiting linear relationship of the considered kernels and Arikan matrix. Further complexity reduction is achieved by identification of common subexpressions. The proposed approach enables SC list decoding of polar codes with some large kernels with lower complexity compared to the codes based on the Arikan kernel with the same performance.

An algorithm for construction of binary polarization kernels of size 16 and 32 with polarization rate greater than 0.5, which admit low complexity processing is proposed. Kernels are obtained by employing such linear transformations of the Arikan matrix, which minimize the complexity of the window processing algorithm, while preserving required rate of polarization. Simulation results show that polar subcodes with obtained kernels can outperform polar codes with Arikan kernel, while having lower decoding complexity.

A decoding algorithm for polar codes with binary 16×16 kernels with polarization rate 0.51828 and scaling exponents 3.346 and 3.450 is presented. The proposed approach exploits the relationship of the considered kernels and the Arikan matrix to significantly reduce the decoding complexity without any performance loss. Simulation results show that polar (sub)codes with 16×16 kernels can outperform polar codes with Arikan kernel, while having lower decoding complexity.

A method for construction of polar subcodes is presented, which aims on minimization of the number of low-weight codewords in the obtained codes, as well as on improved performance under list or sequential decoding. Simulation results are provided, which show that the obtained codes outperform LDPC and turbo codes.