Causal feedback codes for a finite cost-constrained DMC

Let $W(y\mid x)$ be the finite DMC Finite discrete memoryless channels and $c:\mathcal{X}\to [0,\mathrm{\infty })$ the finite input cost from Finite DMC with a nonnegative input cost and feasible budget, with feasible budget $\mathrm{\Gamma }$ from Average-cost fixed-length codes and operational DMC capacity. For an integer $n\ge 1$, a deterministic length-$n$ feedback code with $M\ge 1$ equiprobable messages consists of causal maps $f_t:[M]\times {\mathcal{Y}}^{t-1}\to \mathcal{X}(1\le t\le n)$ and a decoder $g:{\mathcal{Y}}^n\to [M]$. Let $U$ be a uniform random message on $[M]$, and set $X_t=f_t(U,Y^{t-1})$. Given $U=m$, the output law is the product of the row probabilities $W(y_t\mid f_t(m,y^{t-1}))$. Its expected average cost is $\frac{1}{n}{\sum }_{t=1}^n\mathbb{E}[c(X_t)],$ where the expectation is over $U$ and the channel outputs. It is feasible when this quantity is at most $\mathrm{\Gamma }$; its average error is $\mathrm{Pr}[g(Y^n)\mathrm{\ne }U]$ and its rate is $n^{-1}\ln M$ nats per use. The feedback operational capacity $C_{\mathrm{f}\mathrm{b}}(\mathrm{\Gamma })$ is the supremum of rates for which feasible codes for all sufficiently large blocklengths have average error tending to zero and limiting inferior rate at least the stated rate. The no-feedback subclass consists of encoders $f_t$ that do not depend on $y^{t-1}$; it includes every ordinary fixed-length code from Fixed-length codes for a finite DMC. Its rate uses the natural logarithm The natural logarithm as the inverse of the exponential function.