Can Polar Codes (or any other efficient codes) reach the second order capacity?
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.