Can Polar Codes (or any other efficient codes) reach the second order capacity? In channel coding, it is known (e.g. Yury Polyanskiy's thesis, and the arxiv article A Tight Upper Bound for the Third-Order Asymptotics of Discrete Memoryless Channels) that certain codes, for example polar codes, can reach the Shannon limit for large enough block lengths $n$. However, for finite $n$, the best theoretically possible codes are known to satisfy

$$\log M \leq n C - \sqrt{n V}\, \Phi^{-1}(1-ε) + \frac{1}{2} \log n + O(1),$$

where $M$ is the size of the code, $C$ the capacity of the channel, $V$ the dispersion of the channel, and $ε$ the required (average or maximum) error.

Is it known how Polar codes (or any other codes) compare to this fundamental bound? The bound can be achieved using random codes, but I was wondering if there are any efficient codes that come close.
As far as I know, people conjecture that for Polar codes and any fixed DMBSC (discrete memoryless binary symmetric channel), $\log M \leq nC - O(n^{1-c})$ for some absolute constant $c &gt; 0$ and vanishing error probability should be possible (or in other words, in order to be $\epsilon$ close to capacity, only a polynomially large $n$ in $1/\epsilon$ would suffice. However, there is no rigorous published proof available at this point. There are rumors about an upcoming proof, but that's not official yet.