## 3 System Model and Preliminaries
### DKI Coding for the BSC
_where, for every \(\epsilon_{1},\epsilon_{2}>0\), fulfill the bounds \(P_{e,1}(i,\mathsf{K})\leq e_{1}\), \(\forall i\in\mathsf{K}\) and \(P_{e,2}(i,\mathsf{K})\leq e_{2}\), \(\forall i\
otin\mathsf{K}\), where \(\mathsf{K}\in\{\mathsf{K}\subseteq\llbracket M\rrbracket\;;\lvert\mathsf{K} \rvert=K\}\) is an arbitrary subset of \(\llbracket M\rrbracket\) with size \(K\)._  
**Definition 2** ( DKI Coding / Goal Identification Rates ). : _The codebook size \(M(n,R)\) and the goal message set size \(K(n,\kappa)\) are sequences of non-decreasing monotonically functions in the codeword length \(n\), with \(R,\kappa\), and \(l\) indicating the DKI coding rate and the goal identification rate, respectively. In this work, we consider the subsequent functions:_  
\[M(n,R)=2^{nR}\qquad\text{and}\qquad K(n,\kappa)=2^{n\kappa}. \tag{10}\]  
_Thereby, the DKI coding rate, \(R\), and the goal identification rate, \(\kappa\), are defined as follows (additionally, in the literature, other rate definitions for different communication settings are adopted; for example, in the RI [49] problem, the RI coding rate is defined as \(\left(\log\log M\right)/n\), while in the TR [17] or DI [32] problems for a DMC, the TR and DI coding rates are given by \(R=\left(\log M\right)/n\). )_  
\[R=\frac{\log M}{n}\qquad,\qquad\kappa=\frac{\log K}{n}. \tag{11}\]  
**Definition 3** ( Attainable Rate Region ). : _The pair of rates \((R,\kappa)\) is called attainable if, 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. Then, the set of all attainable rate pairs \((R,\kappa)\) is referred to as the attainable rate region for the BSC, \(\mathcal{B}\), and is denoted by \(\mathsf{R}_{\text{DKI}}(\mathcal{B},M,K)\)._  
**Definition 4** ( Capacity Region / Capacity ). : _The operational DKI capacity region of the BSC, \(\mathcal{B}\), is defined as the closure of all attainable rate triples \((R,\kappa)\) (the closure of a set \(\mathsf{A}\) consists of all points in \(\mathsf{A}\) together with all limit points of \(\mathsf{A}\), where the limit point of \(\mathsf{A}\) is a point \(x\) that can be approximated by the points of \(\mathsf{A}\); see [62] for further details), and is denoted by \(\mathbb{C}_{\text{DKI}}(\mathcal{B},M,\mathsf{K})\). For the standard identification (\(K=1\)), the capacity region is specialized to a single point, also called the DI capacity which is the supremum of all attainable DI coding rates, \(R\). The DI capacity is denoted by \(\mathbb{C}_{\text{DI}}(\mathcal{B},M)\)._  
**Remark 1** ( Distribution of Output Statistics ). : _Assuming that the codeword \(\mathbf{c}_{i}\) is sent and the channel output \(\mathbf{y}\) is observed at the receiver, the number of cross-overs (flips) that occurs in the channel is given by \(d_{\mathrm{H}}(\mathbf{y},\mathbf{c}_{i})\). Therefore, the probability that \(k\) cross-overs among the \(n\) channel uses occurs, follows a Binomial distribution with parameters \(n\) and \(\varepsilon\) as follows:_  
\[\Pr\bigl{(}d_{\mathrm{H}}(\mathbf{Y},\mathbf{c}_{i})=k\bigr{)}=\binom{n}{k} \varepsilon^{k}(1-\varepsilon)^{n-k}. \tag{12}\]