## 3 System Model and Preliminaries
### System Model  
We target a communication setting, which is focused on the identification goal; that is, the objective of the decoder is defined as follows: Determine if the sent message belongs to a goal group of messages of size \(K\). In order to do this, the transmitter and the receiver build (the suggested inner and outer bounds on the DKI capacity region functions, whether or not a particular code is utilized for the communication; however, in order to approach the capacity limits, appropriate, explicitly built codes could be needed), a coded communication channel over \(n\), uses of the binary symmetric channel. We assume that the random variables (RVs) \(X\in\{0,1\}\) and \(Y\in\{0,1\}\) indicate model the input and output of the channel. Each binary input symbol is flipped with probability \(0<\varepsilon<1/2\); see Figure 1. The stochastic flipping (the extreme cases of \(\varepsilon=0\) or \(\varepsilon=1/2\) result in \(\mathbb{C}_{\text{TR}}=1\) and \(\mathbb{C}_{\text{TR}}=0\), respectively; hence, these cases are commonly excluded from the analysis) of the input symbol is modeled via an additive binary Bernoulli noise, i.e., \(Z\in\{0,1\}\). Therefore, the input-output relation of channel reads: \(Y=X\oplus Z\), where \(\oplus\) indicate the modulo two addition. That is, the channel input/output \(X/Y\) are related as follows:  
\[W(Y|X)=\left\langle\begin{aligned} 1-\varepsilon& Y=X,\\ \varepsilon& Y\
eq X,\end{aligned}\right. \tag{6}\]  
for all \(X,Y\in\{0,1\}\) and \(0<\varepsilon<1/2\).  
Furthermore, it is assumed that the various channel uses are independent of one another and that the communication channel is memoryless. Therefore, the transition probability distribution for \(n\) channel uses is given by  
\[W^{n}(\mathbf{y}|\mathbf{x})=\prod_{t=1}^{n}W(y_{t}|x_{t})=\varepsilon^{d_{ \text{H}}(\mathbf{x},\mathbf{y})}(1-\varepsilon)^{n-d_{\text{H}}(\mathbf{x}, \mathbf{y})}, \tag{7}\]  
where \(\mathbf{x}=(x_{1},\ldots,x_{n})\) and \(\mathbf{y}=(y_{1},\ldots,y_{n})\) stand for the sent codeword and received signal, respectively, and \(d_{\text{H}}(\cdot)\) denotes the Hamming distance. Observe that \(d_{\text{H}}(\mathbf{x},\mathbf{y})\) is a RV, and follows a Binomial distribution; see Remark 1. We assume that the codewords are restricted by an input constraint of the form \(\frac{1}{n}\sum_{t=1}^{n}x_{t}\leq A\), where \(A>0\) constrain the Hamming weight over the entire \(n\) channel uses in each codeword normalized by the codeword length.  
**Memoryless property:** In the standard modeling of the BSC, we assume that the channel is exploited at different time instances in an independent manner; that is, the communications of symbols at distinct time instances are statistically independent of each other. However, in the physical channels, such as telephone lines with impulse noise or slowly fading radio communications with binary alphabet, communication is usually dispersive and the channel exhibit memory [35; 61]. Therefore, appropriate steps need to be take in order to ensure the orthogonality of the different channel uses. Some immediate approaches include applying interlacing or scrambling the symbols of a codeword; cf. [61] for further details. Therefore, in the analysis, we can assume that such methods can be applied to circumvent the effect of channel memory and assert statistical independence between different channel noise samples to ensure the memoryless property.