Enumeration of Minimum Hamming Weight Polar Codewords with Sublinear Complexity

Abstract

Polar code, with explicit construction and recursive structure, is the latest breakthrough in channel coding field for its low-complexity and theoretically capacity-achieving property. Since polar codes can approach the maximum likelihood performance under successive cancellation list decoding (SCLD), its decoding performance can be evaluated by Bonferroni-type bounds (e.g., union bound) in which the Hamming weight spectrum will be used. Especially, the polar codewords with minimum Hamming weight (PC-MHW) are the most important item in that bound because they make major contributions to the decoding error pattern particularly at high signal-to-noise-ratio. In this work, we propose an efficient strategy for enumerating the PC-MHW and its number. By reviewing the inherent reason that PC-MHW can be generated by SCLD, we obtain some common features of PC-MHW captured by SCLD. Using these features, we introduce a concept of zero-capacity bit-channel to obtain a tight upper bound for the number of PC-MHW, whose computing complexity is sublinear with code length. Furthermore, we prove that the proposed upper bound is really the exact number of PC-MHW in most cases. Guided by the bound and its theoretical analysis, we devise an efficient SCLD-based method to enumerate PC-MHW, which requires less than half of the list size compared with the existing methods.