Abstract: Polar code is the only linear error-correcting channel code that has been proved theoretically to reach the Shannon Limit. Herein, on the basis of the existing studies on the second and third-order kernel matrices of polar codes, the construction criteria of an optimal fourth-order kernel matrix are proposed: the main diagonals are 1, the number of “1” in the last line is 4, and all the matrices conforming to the abovementioned criteria are determined. Unlike the second-order kernel matrix, which only exhibits a single linear form, the fourth-order kernel matrix can take several forms, providing the polarization codes with more options in the construction. Then, taking the kernel matrix as an example, the channel polarization principle is introduced in detail. Finally, the steps for constructing a polar code having a specific block length with a given kernel matrix of any dimension are summarized.

Key words: polar code, kernel matrix, channel coding; polarizability, recursive structure