Research On The Construction Of Kernel Matrices Of Polar Codes Based On Primitive BCH Codes

Polar codes are the first channel coding scheme which that has been proven to reach the Shannon limit in theory,and they were officially incorporated into the 5G standard as a control channel coding scheme in the Enhanced Mobile Broadband(e MBB)scenario in 2016.The kernel matrices are an important component of the polar codes,and the high-dimensional kernel matrices have better polarization effect.BCH codes can be used to construct high-dimensional kernel matrices,but the existing constructions may need to verify the polarization of kernel matrices,or the partial distance may exceed the upper bound.In this paper,we obtain some kernel matrices that don’t need to verify their polarization and satisfy the upper bound of partial distance.Because there are some limitations in the traditional shortening method,and in order to realize the variable code length of polar codes,we improve the traditional shortening method.The main work is as follows:(1)2~n-1-order lower triangular matrices are obtained by means of interception and direct-sum,and then 2~n-dimensional kernel matrices are obtained by extending them.These matrices are lower triangular matrices which satisfy the polarization requirements and don’t need to verify their polarization.A solution is proposed to solve the problem that the partial distance of the kernel matrices exceed their upper bound,the solution guarantees the cyclic structure and sub-lower triangular structure of the2~n-1-order matrices,thereby ensuring the polarization of the2~n-dimensional kernel matrices,and the lower bounds on the exponents of the 2~n-dimensional kernel matrices are obtained.The comparison results show that the lower bounds on the exponents of the 2~n-dimensional kernel matrices constructed in this paper are higher than that of the 2~n-dimensional kernel matrices constructed by Gilbert-Varshamov(G-V)construction,which are improved by 10%to 25%,and the exponent of the 2~n-dimensional kernel matrix constructed in this paper is higher than the previous construction.(2)Because the 2~n-dimensional kernel matrices constructed in this paper have some limitations in the traditional shortening method,and the exponents of the shorted kernel matrices are not high after shortening.This paper improves on the traditional shortening method,and the replacement-1 shorting method is proposed.The replacement-1 shorting method guarantees the shortened kernel matrices are still lower triangle matrices after shorting,which ensures the polarization of the kernel matrices.The comparison results shows that the exponents of the shorted kernel matrices obtained by the replacement-1 shorting method are higher than those obtained by the traditional shortening method,and higher rate of change in the exponents of the kernel matrices obtained with the replacement-1 shorting method than traditional shorting method.