A trellis decoding scheme based on Massey trellis for polar codes with a ternary kernel is proposed. First, we construct trellises for kernel computations of the successive cancellation (SC) decoder based on the Massey trellis. Then, the scheme calculates the bit-channel transition probability through these trellises, which provides an appropriate framework to take advantage of the distributive law and can effectively reduce the number of operations. Simulation results show that for a G5⊗3 polar code of length 243 and rate 0.5, our scheme can save 14.2% of the computational cost with no sacrifice in error performance under the SC decoder.
Polar codes with large kernel matrix have larger polarization rates than polar codes with the original 2×2 kernel matrix, resulting in better decoding error correction performance for the same block length. In this paper, the piecewise Gaussian approximation (PGA) design method based on polar codes with the 2×2 kernel matrix is extended to polar codes with the 3×3 kernel matrix. A recursive formula for the mean of the log-likelihood ratio of the 3×3 kernel matrix polar codes is derived, and the PGA design algorithm for the 3×3 kernel matrix polar codes is given. The simulation results show that the frame error rate performance of polar codes with the 3×3 kernel matrix constructed by the PGA method is better than that the polar codes constructed by the exact Gaussian approximation (EGA) and the approximate Gaussian approximation (AGA) methods.
In this paper, the reciprocal channel approximation (RCA) method is used to design polar codes with large kernels. Firstly, based on the given large kernels, we obtain the encoder graph. Secondly, the bit-channel SNR parameters are successively updated according to the encoder graph by the RCA. Finally, the frozen bits are selected by the SNR of the last bit-channels. The RCA method can avoid the distortion caused by the kernel matrix being too large. Furthermore, in the calculation process, the closed-form expression of the channel capacity is used to avoid excessive use of transcendental functions, which can greatly reduce the calculation cost. The experimental results show that for the G3⊗5 and G5⊗3 with lengths of 243 and 125, and bit rates of 1/2, when the channel SNR is more than 3dB and 2.5dB, respectively, polar codes with large kernels designed by our scheme have better frame error rate than the Gaussian approximation (GA) method.
KEYWORDS: Picture Archiving and Communication System, Forward error correction, Error control coding, Signal to noise ratio, Polarization, Computer simulations, Binary data, Matrices, Convolution, Statistical analysis
Recently, Arikan has proposed a new scheme to transfer the error correction part to external codes for encoding, namely polarization-adjusted convolutional (PAC) coding. However, all existing decoding methods of current research for PAC codes focused on the classic sequential decoding-Fano decoding. The stack decoding which is another well-known sequential decoding, has not been reported. In this paper, we try to do a thorough study of the stack decoding for PAC codes. First, we find that the traditional stack decoding (SD) is not very effective for the PAC codes, especially in terms of complexity. Therefore, we propose a soft output stack decoding (SOSD) for PAC codes, which can reduce the computation very much. In the SOSD algorithm, the discarded paths are recorded and used in the final decision of decoding codeword. The SOSD algorithm can reduce the stack size (amount of computation) very much with almost no decoding performance loss. By carefully choosing the stack size and soft output threshold, in the case of a PAC code with code length 128 and rate 1/2, the SOSD algorithm can reduce 82% of the computation by comparing the conventional SD algorithm.
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