## 3 Information-Theoretic Results
### Information-Theoretic System Model  
As an example, we consider the general system model introduced in [27], which incorporates both common randomness and private randomness (see Figure 2). The encoder and the decoder have access to a shared source of randomness \(J\) uniformly distributed over \([2^{nR_{c}}]\). The encoder and decoder also have private randomness sources \(L_{e}\sim P_{L_{e}}\) and \(L_{d}\) uniformly distributed over \([2^{nR_{d}}]\), respectively. The encoder observes the source \(X^{n}\), \(J,L_{e}\) and selects a message \(M\in[2^{nR}]\). The decoder observes the message, has access to \(J\) and \(L_{d}\), and attempts to reconstruct the input as \(Y^{n}\). [PERSON] et al. [27] showed that the private randomness of the encoder is not useful under the constraint of near-perfect realism. Other variants of the system model include models with (common) side information [43; 46].  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Realism Constraint** & **Definition** & **References** \\ \hline Weak realism & \(\mathcal{D}(\hat{P}_{X^{n}},\hat{P}_{Y^{n}})\leq P\) & [41; 42; 43] \\ Per-symbol realism & \(\mathcal{D}(P_{X^{n}},P_{Y^{n}})\leq P,\ \forall t\in[n]\) & [26] \\ Average marginal realism & \(\sup_{t}\frac{1}{n}\sum_{t=1}^{n}\mathcal{D}(P_{X},P_{Y^{n}})\leq P\) & [9] \\ Strong realism & \(\mathcal{D}(P_{X^{n}},P_{Y^{n}})\leq P\) & [43; 44; 45] \\ Perfect strong realism & \(Y^{n}\sim P_{X^{n}}\) & [25; 31; 39] \\ Near-perfect per-symbol realism & \(\lim_{n\to\infty}\max_{1\leq t\leq n}\mathcal{D}(P_{X},P_{Y_{t}})=0\) & [26; 27] \\ Near-perfect strong realism & \(\lim_{n\to\infty}D(P_{X^{n}},P_{Y^{n}})=0\) & [25; 27] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Taxonomy of realism constraints **(P)**.  
Figure 2: A general system model including common randomness \(J\), encoder private randomness \(L_{e}\), and decoder randomness \(L_{d}\) as in [27].  
**Definition 3**.: _An \((n,2^{nR_{c}},2^{nR_{c}},2^{nR_{d}})\) fixed-length code consists of an encoding function_  
\[f_{n}:\mathcal{X}^{n}\times[2^{nR_{c}}]\times\mathcal{L}_{d}\mapsto[2^{nR}]\qquad \text{(possibly stochastic)}\]  
_and a decoding function_  
\[g_{n}:[2^{nR}]\times[2^{nR_{c}}]\times[2^{nR_{d}}]\mapsto\mathcal{X}^{n}\qquad \text{(possibly stochastic)}.\]  
The system presented here is designed for fixed-length codes. For variable-length codes, the uniform distributions of \(J\) and \(L_{d}\) are replaced with the following constraints:  
\[R\geq\frac{1}{n}H(M),\quad R_{c}\geq\frac{1}{n}H(J),\quad R_{d}\geq\frac{1}{n} H(L_{d}).\]  
In stochastic coding, it is assumed that various degrees of common randomness between the encoder and decoder, as well as private randomness, are available [27]. When common or private randomness is not available, the corresponding random variables \(J\), \(L_{e}\), or \(L_{d}\) are set to constant, and their respective rates become zero. Specifically, when \(J\) and \(L_{e}\) are constant, the encoding function \(f_{n}\) becomes deterministic with respect to \(X^{n}\). Similarly, when \(J\) and \(L_{d}\) are constant, the decoding function \(g_{n}\) becomes deterministic with respect to \(X^{n}\).