A Note on the Mixing Factor of Polar Codes

Over binary-input memoryless symmetric (BMS) channels, the performance of polar codes under successive cancellation list (SCL) decoding can approach maximum likelihood (ML) algorithm when the list size L is greater than or equal to [Formula: see text] , where MF, known as mixing factor of code, represents the number of information bits before the last frozen bit. Recently, Yao et al. showed the upper bound of the mixing factor of decreasing monomial codes with length [Formula: see text] and rate [Formula: see text] when m is an odd number; moreover, this bound is reachable. Herein, we obtain an achievable upper bound in the case of an even number. Further, we propose a new decoding hard-decision rule beyond the last frozen bit of polar codes under BMS channels.