Finite-alphabet point-to-point joint source-channel codes

Let $\mathcal{S},\mathcal{X},\mathcal{Y},\widehat{\mathcal{S}}$ be nonempty finite alphabets, let $P_S$ be a probability vector on $\mathcal{S}$, and let $W$ be a finite discrete memoryless channel from $\mathcal{X}$ to $\mathcal{Y}$ as in Finite discrete memoryless channels. For positive integers $k,n$, a deterministic $(k,n)$ joint source-channel code is a pair of maps $f_k:{\mathcal{S}}^k\to {\mathcal{X}}^n,g_k:{\mathcal{Y}}^n\to {\widehat{\mathcal{S}}}^k\mathrm{.}$ The source block has law $P_S^{\otimes k}$, the channel output conditional on input $x^n$ has law $W^n(\cdot \mid x^n)$, and the reproduction is ${\widehat{S}}^k=g_k(Y^n)$. The code has no feedback.

A code sequence at channel-use ratio $\rho$ consists of such codes with source lengths $k\to \mathrm{\infty }$, channel lengths $n_k$, and $n_k\mathrm{/}k\to \rho \in (0,\mathrm{\infty })$. In the lossless case $\widehat{\mathcal{S}}=\mathcal{S}$, its block error is $\mathrm{Pr}[{\widehat{S}}^k\mathrm{\ne }{S}^k]$. In the lossy case, for finite nonnegative single-letter distortion $d$, its normalized block distortion is $d_k$ from Single-letter and block distortion; expected distortion and the positive-slack excess-distortion probabilities $\mathrm{Pr}[d_k>D+\epsilon ]$ for every $\epsilon >0$ are distinct criteria.