On the scaling of Polar Codes: II. The behavior of un-polarized channels

2010/02/17 by S. Hamed Hassani, Hassani, S. Hamed, Kasra Alishahi +4
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Cellular Automata and Applications #DNA and Biological Computing #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT

paper · pdf · doi:10.48550/arxiv.1002.3187

Submitted to ISIT 2010

openalex publication_date 2010/02/17 · arxiv created 2010/02/18 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We provide upper and lower bounds on the escape rate of the Bhattacharyya process corresponding to polar codes and transmission over the the binary erasure channel. More precisely, we bound the exponent of the number of sub-channels whose Bhattacharyya constant falls in a fixed interval [a,b]. Mathematically this can be stated as bounding the limit lim_{n → ∞} (1)/(n) ln ℙ(Z_n ∈ [a,b]), where Z_n is the Bhattacharyya process. The quantity ℙ(Z_n ∈ [a,b]) represents the fraction of sub-channels that are still un-polarized at time n.

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