This is a small correction to the proof of Lemma 11 in [1]. Many thanks to Ishaque Ashar for discovering the error.

Erroneous statement

The erroneous statement is after Eq. (79) in the proof of Lemma 11. We claimed that the Markov chain Z1:i11:nCWiZi1:n holds. However, looking at the dependencies introduced through the chaining in Fig. 3, we see that the correct Markov chain is instead Zi1:nCWiZi+1:bn1:n.

Fixing the error

Fortunately, there is an easy correction. By Pinsker’s inequality,

V(p~Z1:bn1:n,i=1bnp~Zi1:n)D(p~Z1:bn1:ni=1bnp~Zi1:n).

The relative entropy can then be bounded as

(1)D(p~Z1:bn1:ni=1bnp~Zi1:n)=i=1bnD(p~Zi1:n|Zi+1:bn1:np~Zi1:n|p~Zi+1:bn1:n)(2)=i=1bnI(Z~i1:n;Z~i+1:bn1:n)(3)i=1bnI(Z~i1:n;CWiZ~i+1:bn1:n)(4)=i=1bn[I(Z~i1:n;CWi)+I(Z~i1:n;Z~i+1:bn1:n|CWi)=0](5)=i=1bnI(Z~i1:n;CWi)(6)bnδn(5)

Therefore, V(p~Z1:bn1:n,i=1bnp~Zi1:n)bnδn(5)bnδn(5).

References

  1. G. Frèche, M. Bloch, and M. Barret, “Polar Codes for Covert Communications over Asynchronous Discrete Memoryless Channels,” Entropy, vol. 20, no. 1, p. 3, Dec. 2017.