Finite-alphabet multiterminal lossy source code

Fix $k\ge 2$ and a finite IID source vector $(X_{1,t},\dots ,X_{k,t}{)}_{t\ge 1}$ with law $P_{X_1\cdots X_k}$, as in Finite-alphabet IID multisource. For each $i$, let ${\widehat{\mathcal{X}}}_i$ be a finite nonempty reproduction alphabet and let $d_i:{\mathcal{X}}_i\times {\widehat{\mathcal{X}}}_i\to [0,d_{i,\max }]$ be finite and bounded.

A length-$n$ fixed-length multiterminal lossy code consists of separate deterministic encoders $f_{i,n}:{\mathcal{X}}_i^n\to [M_i](1\le i\le k)$ and one deterministic common decoder $g_n:{\prod }_{i=1}^k[M_i]\to {\prod }_{i=1}^k{\widehat{\mathcal{X}}}_i^n\mathrm{.}$ Write ${\widehat{X}}_i^n$ for the $i$th reconstruction and define its expected per-letter distortion by ${\mathrm{\Delta }}_{i,n}:=\mathbb{E}[n^{-1}{\sum }_{t=1}^n{d}_i(X_{i,t},{\widehat{X}}_{i,t})]$. Encoder $i$ has rate $n^{-1}\ln M_i$ nats per source symbol. A vector $(R_1,\dots ,R_k)$ is achievable at distortions $(D_1,\dots ,D_k)$ when there is an increasing sequence of blocklengths with each rate limsup at most $R_i$ and each distortion limsup at most $D_i$. The operational rate-distortion region is the closure of these achievable vectors.

No encoder observes another encoder's source block. The common decoder receives all message indices and outputs one reconstruction sequence for each source.