Finite fixed-length multiterminal lossless code

Fix a finite-alphabet IID multisource from Finite-alphabet IID multisource and a blocklength $n\ge 1$. Choose positive integers $M_1,\dots ,M_k$. A fixed-length multiterminal lossless code consists of separate deterministic encoders $f_{i,n}:{\mathcal{X}}_i^n\to [M_i]:=\{1,\dots ,M_i\},1\le i\le k,$ and one common deterministic decoder $g_n:{\prod }_{i=1}^k[M_i]\to {\prod }_{i=1}^k{\mathcal{X}}_i^n\mathrm{.}$ Its block-error probability is $P_e^{(n)}:=\mathbb{P}\text{\hspace{0.17em}⁣}\left[g_n\right(f_{1,n}(X_1^n),\dots ,f_{k,n}(X_k^n))\mathrm{\ne }(X_1^n,\dots ,X_k^n)]\mathrm{.}$ The rate of encoder $i$ is $R_{i,n}:=n^{-1}\ln M_i$ nats per source symbol, with the logarithm convention of Information units and logarithm-base convention. A vector $R=(R_1,\dots ,R_k)\in [0,\mathrm{\infty }{)}^k$ is achievable if there are strictly increasing blocklengths $n_j\to \mathrm{\infty }$ and such codes for which $P_e^{(n_j)}\to 0$ and ${\mathrm{lim\hspace{0.17em}sup}}_{j\to \mathrm{\infty }}{R}_{i,n_j}\le R_i$ for every $i$. The encoders do not observe other terminals' source blocks; only the common decoder receives all message indices.

Each index alphabet can be transmitted using $\lceil {\log }_2{M}_i\rceil$ bits of fixed length. The corresponding rate in nats differs from $n^{-1}\ln M_i$ by at most $(\ln 2)\mathrm{/}n$ due to integer bit rounding.