Abstract
The polar codes defined by the kernel matrix are a class of codes with low coding-decoding complexity and can achieve the Shannon limit. In this paper, a novel method to construct the 2n-dimensional kernel matrix is proposed, that is based on primitive BCH codes that make use of the interception, the direct sum and adding a row and a column. For ensuring polarization of the kernel matrix, a solution is also put forward when the partial distances of the constructed kernel matrix exceed their upper bound. And the lower bound of exponent of the 2n-dimensional kernel matrix is obtained. The lower bound of exponent of our constructed kernel matrix is tighter than Gilbert-Varshamov (G-V) type, and the scaling exponent is better in the case of 16-dimensional.
Highlights
Polar codes can achieve the Shannon limit for binary-input discrete memoryless channels (BI-DMC) in theory and with low encoding and decoding complexity [1]
The lower bound of exponent of our constructed kernel matrix is tighter than Gilbert-Varshamov (G-V) type, and the scaling exponent is better in the case of 16-dimensional
In order to construct a kernel matrix with larger exponent, the sub-matrices of each layer should be intercepted from different generator matrices, because as the number of primitive BCH code error corrections increases, the minimum distance of these sub-matrices increases, the partial distance of the 2n -dimensional kernel matrix will increases, the exponent will increases
Summary
Polar codes can achieve the Shannon limit for binary-input discrete memoryless channels (BI-DMC) in theory and with low encoding and decoding complexity [1]. Korada [5] proposed to construct a 2n −1 -dimensional kernel matrix with BCH codes, which increased the dimension of kernel matrix. E. Moskovskaya et al [6], based on [5] [7], put forward a method to construct a 2n -dimensional kernel matrix with extended BCH codes, which further increased the dimension of kernel matrix. Based on [5] [6], we take advantage of the primitive BCH codes to design a higher-dimensional kernel matrix which meets the upper bound of partial distance. Compared with the work in [5], the proposed construction enjoys two advantages, one is that the kernel matrix has a higher dimension, and the other is that the obtained kernel matrix is naturally lower triangular, guaranteeing the polarization property of kernel matrix. The comparison result shows that the proposed 2n -dimensional kernel matrix has a tighter lower bound of the exponent than the GilbertVarshamov (G-V) type construction in [5], the scaling exponent is not very different from [6], and 16-dimensional kernel matrix is even slightly better than [6]
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