Causal feedback codes for a finite-dimensional Gaussian channel

Assume AC The Axiom of Choice and use the channel in Finite-dimensional additive Gaussian channel with average power with dimension $d$ and budget $P$. A length-$n$ code with $M\ge 1$ equiprobable messages is a tuple of Borel encoders $f_t:[M]\times ({\mathbb{R}}^d{)}^{t-1}\to {\mathbb{R}}^d,1\le t\le n,$ and a Borel decoder $g:({\mathbb{R}}^d{)}^n\to [M]$. At time $t$ the input is $X_t=f_t(M,Y^{t-1})$ and the output is $Y_t=X_t+Z_t$, with the fresh independent Gaussian noise of the channel definition. The code meets the expected average-power constraint when $\frac{1}{n}\mathbb{E}\left[{\sum }_{t=1}^n\mathrm{\parallel }{X}_t{\mathrm{\parallel }}_2^2\right]\le P\mathrm{.}$ Its average error is $\mathrm{Pr}[g(Y^n)\mathrm{\ne }M]$ and its rate is $n^{-1}\ln M$ nats per vector use. If every $f_t$ is independent of $Y^{t-1}$, the code does not use feedback. Define $C_{\mathrm{f}\mathrm{b}}(P)$ as the supremum of rates $R$ for which feasible feedback codes for all sufficiently large $n$ have average error tending to zero and $\mathrm{lim\hspace{0.17em}inf}{n}^{-1}\ln M_n\ge R$. Define $C_{\mathrm{n}\mathrm{f}}(P)$ by restricting to codes that do not use feedback. Both capacities use natural-log nats per vector use The natural logarithm as the inverse of the exponential function.