Finite-Length Scaling for Polar Codes
Seyed Hamed Hassani, Kasra Alishahi, Rudiger L. Urbanke · IEEE Transactions on Information Theory · 2014
Consider a binary-input memoryless output-symmetric channel\(W\). Such a channel has a capacity, call it\(I(W)\), and for any\(R0\), then the required block-length\(N\)scales in terms of the rate\(R < I(W)\)as\(N \geq {\alpha }/{(I(W)-R)^{\underline {\mu }}}\), where\(\alpha \)is a positive constant that depends on\(P_{\rm e}\)and\(I(W)\). We show that\(\underline {\mu } = 3.579\)is a valid choice, and we conjecture that indeed the value of\(\underline {\mu }\)can be improved to\(\underline {\mu }=3.627\), the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like\(N \leq {\beta }/{(I(W)-R)^{\overline {\mu }}}\), where\(\beta \)is a constant that depends on\(P_{\rm e}\)and\(I(W)\), and\(\overline {\mu }=6\).