## 4 DKI Capacity Region of the BSC
### Inner Bound (Achievability Proof)
In addition, we exploit some bound on the Binomial CDF. Moreover, the error exponents yield the feasible range for the goal identification rate \(\kappa\). Before we start the analysis, we introduce the subsequent parameter definitions and conventions: Fix \(\epsilon_{1},\epsilon_{2}>0\) and let \(\zeta_{0},\zeta_{1}>0\) be arbitrarily small constants. Further, let introduce the subsequent conventions:  
* \(Y_{t}(i)\) is output of channel at time \(t\)_conditioned_ that \(\mathbf{x}=\mathbf{c}_{i}\), i.e., \(\gamma_{t}(i)=\mathbf{c}_{i,t}\oplus Z_{t}\). * The vector of symbols is \(\mathbf{Y}(i)\triangleq(\gamma_{1}(i),\ldots,Y_{n}(i))\). * **Sort I errors:** This error event occur when the transmitter sends \(\mathbf{c}_{i}\), yet \(\mathbf{y}\
otin\mathcal{T}_{\mathsf{K}}\) for every \(i\in\mathsf{K}\). More specifically, the sort I error probability is given by \[\frac{1}{n}\sum_{t=1}^{n}c_{t}\leq A. \tag{29}\]0 (54)  
In order to show that the probability term provided in (54) tends to zero for asymptotic codeword lengths, we show that this term is upper bounded by certain upper tail of the Binomial CDF. Next, employing existing bounds for this tail given in Appendix A4, we establish an upper bound on such an upper tail which vanishes in the asymptotic. The extensive analysis for the sort I errors is provided in Appendix A. **Sort II errors:** The sort II error event happens when \(\mathbf{Y}(i)\in\mathbb{T}_{K}\) while the transmitter sent \(\mathbf{c}_{i}\) with \(i\
otin\mathbb{K}\). Then, for each possible \(\binom{M}{K}\) case of \(\mathbb{K}\), where \(i\
otin\mathbb{K}\), the sort II error probability is given by  
\[\frac{1}{n}\sum_{t=1}^{n}c_{t}\leq A. \tag{29}\]1  
To show that the probability term provided in (55) vanishes for asymptotic regime, we break this term into two new terms and address them separately. One of the terms is shown to vanish by exploiting the proof derived in the sort I error analysis. For the other term, using standard techniques we show that it corresponds to certain Binomial CDF. Then, employing some existing bounds on such Binomial CDF given in Appendix A.5, we assert an upper bound for it which tends to zero in the asymptotic. The detailed analysis for the sort II errors is provided in Appendix B. Observe that considering the established lower bound on the DKI coding rate \(R\) and the established upper bound on the goal identification rate \(\kappa\), as provided in (41) and (48) and (49), means that we have shown for every \(e_{1},e_{2}>0\) and sufficiently large \(n\), there exists an \((n,M(n,R),K(n,\kappa),e_{1},e_{2})\)-BSC-DKI code, such that the set \(\mathsf{R}_{\text{DKI}}(\mathcal{B},M,K)\) of all attainable rate pairs \((R,\kappa)\) contains  
\[\frac{1}{n}\sum_{t=1}^{n}c_{t}\leq A. \tag{29}\]2  
with  
\[\mathsf{R}^{\text{inn}}_{\beta}\triangleq\left\langle \begin{array}{ll}\{( R,\kappa);\,0\leq R\leq H(A)-H(\beta),\,0\leq\kappa<\min(\kappa^{1}_{\text{UB}},\, \kappa^{2}_{\text{UB}})\}&\text{if $A<1/2$},\\ \{(R,\kappa);\,0\leq R\leq 1-H(\beta),\,0\leq\kappa<\min(\kappa^{1}_{\text{UB}},\, \kappa^{2}_{\text{UB}})\}&\text{if $A\geq 1/2$},\end{array} \right. \tag{57}\]  
where \(\kappa^{1}_{\text{UB}}\) and \(\kappa^{2}_{\text{UB}}\) are provided in (A.58) and (A.59), respectively. **Remark 3**. : _Methodology for establishing the feasible region of \(\beta\). Observe that, since the parameter \(\beta\) adjusts the radius of the hyper spheres used in the codebook construction, a trivial restriction on it would be as follows: \(\beta\geq 0\). Next, employing the Hamming distance property of Lemma 1 and Lemma 2, \(\beta\) can not be greater or equal than 1; therefore, we conclude that \(0\leq\beta<1\). Now, we exclude the boundary points \(\beta=0\), since it makes the upper bounds on the \(\kappa\) equal to zero (\(\kappa<0\)), which is a contradiction since \(\kappa\geq 0\). Next, we focus on the arguments of \(T_{\varepsilon}(\cdot)\) and \(H(\cdot)\) given in (A.58) and (A.59); see Figure 7. First, observe that the function \(f_{2}(\varepsilon,\beta)\) (cf.