混合多核极化码指数的收敛性

Convergence of Hybrid Multi-Kernel Polar Codes Exponent

摘要: 混合多核极化码存在极化现象且有良好的极化性能。极化码对应的极化矩阵的指数是度量极化码性能的重要指标，因此本文主要研究混合多核极化码的指数。本文将多核极化码部分距离的表达式进行拓展，得到混合多核极化码部分距离，根据极化码部分距离和指数的关系推导出了混合多核极化码指数的表达式，即混合多核极化码指数可以表示为其组成矩阵的线性组合，并得到了混合多核极化码指数的上下界；然后基于混合多核极化码的极化现象，提出了一种特殊的混合多核极化码的矩阵选择规则，证明了在这种矩阵选择规则下产生的混合多核极化码的指数是收敛且与信道容量相关；最后应用本文得出的定理，对一些已有的实验图像和猜想进行了解释。

Abstract: Hybrid multi-kernel polar codes are polarized and have good polarization performance. The expo-nent of the polarization matrix corresponding to the polar code is an important metric to measure the performance of the polar codes, so this paper focuses on the exponent of the hybrid multi-kernel polar codes. In this paper, we extend the expression of partial distance of multi-kernel polar codes to obtain the expression of partial distance of hybrid multi-kernel polar codes, and the expression of exponent of hybrid multi-kernel polar codes is derived based on the relationship between partial distance and exponent of polar codes, which can be expressed as a linear combination of its constit-uent matrices, and the upper and lower bounds of exponent of hybrid multi-kernel polar code are obtained. Then, based on the polarization phenomenon of hybrid multi-kernel polar codes, a special matrix selection rule for hybrid multi-kernel polar codes is proposed, and it is proved that the ex-ponent of hybrid multi-kernel polar codes generated under this matrix selection rule is convergent and channel capacity dependent. Finally the theorems derived in this paper are applied to explain some existing experimental images and conjectures.