Universal Bounds on the Scaling Behavior of Polar Codes

We consider the problem of determining the trade-off between the rate and the block-length of polar codes for a given block error probability when we use the successive cancellation decoder. We take the sum of the Bhattacharyya parameters as a proxy for the block error probability, and show that there exists a universal parameter $μ$ such that for any binary memoryless symmetric channel $W$ with capacity $I(W)$, reliable communication requires rates that satisfy $R< I(W)-αN^{-\frac{1}μ}$, where $α$ is a positive constant and $N$ is the block-length. We provide lower bounds on $μ$, namely $μ\geq 3.553$, and we conjecture that indeed $μ=3.627$, the parameter for the binary erasure channel.

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