Corrections to “Abelian Group Codes for Channel Coding and Source Coding” [May 15 2399-2414]
S. Sandeep Pradhan, Mohsen Heidari, Aria G. Sahebi · IEEE Transactions on Information Theory · 2018
The group capacity of a discrete memoryless channel$(\mathcal {X},\mathcal {Y},W_{Y|X})$is characterized with maximal probability of error in of[1, Sec. II]. There is a mistake in the proof of achievability as given in Section VII.A. It is correctly shown on page 2408–2409 that\begin{equation*} \lim _{n \rightarrow \infty } \max _{a} \mathbb {E} \left [{ P(E(a)) }\right ] =0 \end{equation*}if for all$\hat {\theta } eq \boldsymbol {s}$,\begin{align*}&\hspace {-0.5pc}R \frac {\sum _{(p,s)\in \mathcal {S}(G)} (s- \hat {\theta }_{p,s}) w_{p,s} \log p}{\sum _{(p,s) \in \mathcal {S} (G)} s w_{p,s} \log q} \\&\qquad \qquad \quad \qquad <\log |H_{\eta ^{*}+\hat {\theta }}|-H(X_{\eta ^{*},b}|Y,[X_{\eta ^{*},b}]_{\hat {\theta }}) - O(\epsilon ). \end{align*}However it is incorrectly claimed that the achievability conditions are: for all$\hat {\theta } eq \pmb {s}$,\begin{equation*} R\le \frac {1}{1-\omega _{\hat {\theta }}} I(X_{\eta ^{*},b};Y|[X_{\eta ^{*},b}]_{\hat {\theta }}). \end{equation*}Our original objective was to characterize the average error group capacity of a discrete memoryless channel. The average error is more widely used than the maximal error. Although we had the proof of achievability for the average error case, we could not prove the converse. So we settled for characterizing the maximal error group capacity. In light of the above error, we have the following resolution.