## 4 DKI Capacity Region of the BSC
### Upper Bound (Converse Proof)  
Before we start the converse proof, some comprehensive steps are explained: We show that the feasible input set (subset of the input sequences that fulfills the Hamming constraint) can be entirely exhausted for selection of the codewords. To this end, we establish an one-to-one mapping between the message and input sets. Hence, the number of messages \(2^{\kappa R}\) is bounded by the size of the feasible input set. More specifically, depending on whether or not an effective Hamming weight constraint is imposed on the input of the channel, we divide it into two cases and address them separately. In particular, the converse proof for each case consists of the subsequent two main technical steps. \(\diamondsuit\)Step 1: we show in Lemma 3 that for any attainable DKI rate whose error probabilities of sort I and sort II tends to zero as \(n\to\infty\)), any pair of distinct messages are associated with different codewords. \(\diamondsuit\)Step 2: exploiting Lemma 3, we acquire an upper bound for the DKI codebook size of a the BSC. Figure 7: Depiction of the error exponent for a BSC. The tangent line of the binary entropy function \(H(p)\) in the cross-over probability point \(0<p=\varepsilon<1/2\), calculated for \(\varepsilon<p=\delta<(1-\beta)\,\varepsilon+\beta/2\), marked in green, is denoted by \(T_{\_}{\varepsilon}(\delta)\). For a given cross-over probability \(\varepsilon\), the difference between \(T_{\_}{\varepsilon}(\delta)\) and \(H(\delta)\) is referred to as the error exponent. For example, the upper bounds on the goal identification rate \(\kappa\) calculated in (111) and (112) are two different error exponents that are derived in the sort II error analysis. The minimum of these error exponents is the bottleneck for the rate \(\kappa\), i.e., an eligible upper bound. We begin with the below lemma on a DKI codebook size. Lemma 3 ( DKI codebook size ). : _Consider a sequence of \((n,M(n,R),K(n,\kappa),e_{1}^{(n)},e_{2}^{(n)})\)-BSC-DKI codes \((\mathcal{C}^{(n)},\,\mathcal{T}^{(n)})\), such that \(e_{1}^{(n)}\) and \(e_{2}^{(n)}\) tend to zero as \(n\to\infty\). Then, given a sufficiently large \(n\), the codebook \(\mathcal{C}^{(n)}\) satisfies the subsequent property: two different messages \(i_{1},i_{2}\in\llbracket M\rrbracket\) cannot have the same codeword representing them; that is,_  
\[i_{1}\
eq i_{2}\quad\Rightarrow\quad\mathbf{c}_{i_{1}}\
eq\mathbf{c}_{i_{2}}. \tag{58}\]  
Proof. : Contrarily, suppose that there are two messages \(i_{1}\) and \(i_{2}\), such that \(i_{1}\
eq i_{2}\), and  
\[\mathbf{c}_{i_{1}}=\mathbf{c}_{i_{1}}=x^{n}, \tag{59}\]  
for some \(x^{n}\in\mathcal{X}^{n}\). Since \((\mathcal{C}^{(n)},\,\mathcal{T}^{(n)})\) forms a \((n,M(n,R),K(n,\kappa),e_{1}^{(n)},e_{2}^{(n)})\)-BSC-DKI code, as stated in Definition 1, it implies that for every possible choice (arrangement) of the goal message set \(\mathbb{K}\subseteq\llbracket M\rrbracket\) of size \(K\), the upper bound on the sort I and sort II error probabilities, i.e., \(\varepsilon_{1}^{(n)}\) and \(e_{2}^{(n)}\), respectively, tends to zero as \(n\) tends to infinity. Remark 6.: _Decoder in converse proof_. While we imposed a concrete structure on the decoding set \(\mathcal{T}_{\mathbb{K}}\), in the achievability proof provided in Section 4.2, i.e., we set \(\mathcal{T}_{\mathbb{K}}=\bigcup_{i_{1}\in\mathbb{K}}\,\overline{v}_{i_{1}}\), the converse proof treats the decoding set \(\mathcal{T}_{\mathbb{K}}\) as a generic function. Next, we review the definition of a BSC DKI code found in (1), and concentrate on the underlying presumptions about the characteristics of a particular series of BSC DKI codes \((\mathcal{C}^{(n)},\,\mathcal{T}^{(n)})\) found in Lemma 3. The subsequent property is endowed by such a code sequence with five parameters, \((n,M(n,R),K(n,\kappa),e_{1}^{(n)},e_{2}^{(n)})\).