Finite-dimensional additive Gaussian channel with average power

Assume the Axiom of Choice The Axiom of Choice. Fix an integer $d\ge 1$ and a real symmetric positive-definite matrix $K\in {\mathbb{R}}^{d\times d}$. The $d$-dimensional additive Gaussian channel with power budget $P\ge 0$ has input and output spaces ${\mathbb{R}}^d$ and single-use law $Y=X+Z,Z\sim N_d(0,K),Z\perp X\mathrm{.}$ At each use a fresh independent copy of $Z$ is added, so the channel is memoryless. The input cost is $c(x)=\mathrm{\parallel }x{\mathrm{\parallel }}_2^2$. A length-$n$ deterministic code with $M$ equiprobable messages meets the average block power constraint $P$ when $\frac{1}{nM}{\sum }_{m=1}^M{\sum }_{t=1}^n\mathrm{\parallel }{x}_t(m){\mathrm{\parallel }}_2^2\le P\mathrm{.}$ Rates are measured in nats per vector use. The noise law is Multivariate normal law, including singular covariance; positive definiteness of $K$ gives the Lebesgue density stated in A positive-definite multivariate normal law has its Gaussian density. The scalar case $d=1$ is the real AWGN channel. Rates use the natural logarithm The natural logarithm as the inverse of the exponential function.