Binary nonlinear kernels of maximum exponents of polar codes of dimensions up to sixteen

Polar codes proposed by Arikan are based on a linear kernel of dimension two with exponent 0.5. In this paper, binary kernels of maximum exponents of dimensions up to 16 are presented except for the case of dimension 12 where the maximum exponent is shown to be attained by either a constructed linear kernel or a possible nonlinear kernel with a specified partial distance sequence. The results show that the minimum dimension for which there exists a kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arikan, is 14. For dimensions 14, 15, 16, discussed by Presman et al., along with 13, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.