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^-1μ, where $α$ is a positive constant and N is the block-length. We provide lower bounds on $μ$, namely μ≥ 3.553, and we conjecture that indeed $μ=3.627$, the parameter for the binary erasure channel.