Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors

Consider the transmission of a polar code of block length
$N$
and rate
$R$
over a binary memoryless symmetric channel
$W$
and let
$P_{\mathrm{ e}}$
be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters
$R$
,
$N$
,
$P_{\mathrm{ e}}$
, and the quality of the channel
$W$
quantified by its capacity
$I(W)$
and its Bhattacharyya parameter
$Z(W)$
. In previous work, two main regimes were studied. In the
error exponent
regime, the channel
$W$
and the rate
$R<I(W)$
are fixed, and it was proved that the error probability
$P_{\mathrm{ e}}$
scales roughly as
$2^{-\sqrt {N}}$
. In the
scaling exponent
approach, the channel
$W$
and the error probability
$P_{\mathrm{ e}}$
are fixed and it was proved that the gap to capacity
$I(W)-R$
scales as
$N^{-1/\mu }$
. Here,
$\mu $
is called
scaling exponent
and this scaling exponent depends on the channel
$W$
. A heuristic computation for the binary erasure channel (BEC) gives
$\mu =3.627$
and it was shown that, for any channel
$W$
,
$3.579 \le \mu \le 5.702$
. Our contributions are as follows. First, we provide the tighter upper bound
$\mu \le 4.714$
valid for any
$W$
. With the same technique, we obtain the upper bound
$\mu \le 3.639$
for the case of the BEC; this upper bound approaches very closely the heuristically derived value for the scaling exponent of the erasure channel. Second, we develop a trade-off between the gap to capacity
$I(W)-R$
and the error probability
$P_{\mathrm{ e}}$
as the functions of the block length
$N$
. In other words, we neither fix the gap to capacity (error exponent regime) nor the error probability (scaling exponent regime), but we do consider a
moderate deviations
regime in which we study how fast both quantities, as the functions of the block length
$N$
, simultaneously go to 0. Third, we prove that polar codes are not affected by
error floors
. To do so, we fix a polar code of block length
$N$
and rate
$R$
. Then, we vary the channel
$W$
and study the impact of this variation on the error probability. We show that the error probability
$P_{\mathrm{ e}}$
scales as the Bhattacharyya parameter
$Z(W)$
raised to a power that scales roughly like
${\sqrt {N}}$
. This agrees with the scaling in the error exponent regime.