## 4 DKI Capacity Region of the BSC
### Main Results  
Our DKI capacity region theorem for the BSC, \(\mathcal{B}\), is stated below. **Theorem 1**. : _Let \(  
_Next, assume that \(\mathcal{B}\) endows an exponential size for the codebook and the goal message set, i.e., \(M(n,R)=2^{nR}\) and \(K(n,\kappa)=2^{n\kappa}\), respectively, where the codewords are subject to the Hamming weight constraint of the form \(n^{-1}\sum_{t=1}^{n}c_{i,t}\leq A,\forall i\in[\![M]\!]\). Now, let us define the subsequent functions:_  
\[f_{1}(\varepsilon,\beta)\triangleq \frac{(1-\beta/2)\varepsilon-\beta/4}{1-\beta}, \tag{13}\] \[f_{2}(\varepsilon,\beta)\triangleq (1-\beta/2)\varepsilon+\beta/4. \tag{14}\]  
_Next, let us define the inner and outer rate regions, i.e., \(\mathsf{R}^{\rm{im}}(\mathcal{B})\) and \(\mathsf{R}^{\rm{out}}(\mathcal{B})\), respectively, as follows:_  
\[\mathsf{R}^{\rm{imn}}(\mathcal{B})\triangleq\bigcup_{\beta\in(0,\beta_{\rm{ max}})}\mathsf{R}^{\rm{imn}}_{\beta}(\mathcal{B}), \tag{15}\]  
_where_  
\[\mathsf{R}^{\rm{im}}_{\beta}(\mathcal{B})\triangleq\left\langle \begin{array} []{ll}\{(R,\kappa);&0\leq R\leq H(A)-H(\beta),&0\leq\kappa<\min(\kappa^{1}_{ \rm{UB}},\,\kappa^{2}_{\rm{UB}})\}&A<1/2,\\ \{(R,\kappa);&0\leq R\leq 1-H(\beta),&0\leq\kappa<\min(\kappa^{1}_{\rm{UB}}, \,\kappa^{2}_{\rm{UB}})\}&A\geq 1/2,\end{array} \right. \tag{16}\]  
_with_  
\[\kappa^{1}_{\rm{UB}} \triangleq T_{\rm{e}}(f_{1}(\varepsilon,\beta))-H(f_{1}( \varepsilon,\beta)), \tag{17}\] \[\kappa^{2}_{\rm{UB}} \triangleq T_{\rm{e}}(f_{2}(\varepsilon,\beta))-H(f_{2}( \varepsilon,\beta)), \tag{18}\]  
_and_  
\[\mathsf{R}^{\rm{out}}(\mathcal{B})\triangleq\left\langle \begin{array}{ll}\{ (R,\kappa);&0\leq R\leq H(A),&0\leq\kappa\leq H(A)\}&A<1/2,\\ \{(R,\kappa);&0\leq R\leq 1,&0\leq\kappa\leq 1\}&A\geq 1/2.\end{array} \right. \tag{19}\]  
_Then, the DKI capacity region \(\mathbb{C}_{\rm{DKI}}(\mathcal{B},M,K)\) is bounded by_  
\[\mathsf{R}^{\rm{in}}(\mathcal{B})\subseteq\mathbb{C}_{\rm{DKI}}(\mathcal{B},M, K)\subseteq\mathsf{R}^{\rm{out}}(\mathcal{B}). \tag{20}\]  
**Proof of Theorem 1. ** The proof of Theorem 1 comprises two components, presented in Sections 4.2 and 4.3, respectively, which are the inner and the outer bound proofs. \(\Box\)  
**Corollary 1** ( DI Capacity of The BSC ). : _The inner and outer bounds for the DKI capacity region of the BSC, \(\mathcal{B}\), for an extreme case (standard identification) where the goal message set consists of only one message, i.e., \(K=1\), recover the previous results for the BSC with Hamming constraint ([32] Ex. 1):_  
\[\mathbb{C}_{\rm{DI}}(\mathcal{B},M)=\left\{ \begin{aligned} & H(A)&\text{if }A<1/2,\\ & 1&\text{if }A\geq 1/2,\end{aligned} \right. \tag{21}\]  
_and the BSC without Hamming constraint ([33] Th. 3.1):_  
\[\mathbb{C}_{\rm{DI}}(\mathcal{B},M)=1. \tag{22}\]  
**Proof. ** The proof is obtained directly by placing \(K=1\) into the upper bounds given in (17) and (18) in Theorem 1, and making further mathematical simplifications. In particular, we show that closure of the inner bound for \(K=1\) coincides the outer bound. Therefore, a full characterization of the capacity region is yielded. We begin with the subsequent observation: The upper bounds provided in (17) and (18) for \(K=2^{n\kappa}=1\left(\kappa=0\right)\) tend to zero. That is,  
\[\left.\mathsf{R}^{\text{inn}}(\mathcal{B})\right|_{\kappa=0}=\bigcup_{\beta\in \left(0,\beta_{\max}\right)}\left.\mathsf{R}^{\text{inn}}_{\beta}(\mathcal{B}) \right|_{\kappa=0} \tag{23}\]  
where  
\[\left.\mathsf{R}^{\text{inn}}_{\beta}(\mathcal{B})\right|_{\kappa=0}=\left\langle \begin{array}{ll}\left\{\left(R,\kappa\right);\,0\leq R\leq H(A)-H(\beta), \,\kappa=0\right\}&\text{if }A<1/2,\\ \left\{\left(R,\kappa\right);\,0\leq R\leq 1-H(\beta),\,\kappa=0\right\}& \text{if }A\geq 1/2.\end{array} \right. \tag{24}\]  
Next, observe that the outer bound provided in (19) for \(K=2^{\text{m\kappa}}=1\left(\kappa=0\right)\) is given by  
\[\left.\mathsf{R}^{\text{out}}(\mathcal{B})\right|_{\kappa=0}\triangleq\left\langle