### Binary Symmetric Channel  
A binary symmetric channel (BSC) in information/coding theory is one of the most well-known and fundamental models for communications channels where the input and output alphabets are binary, i.e., \(\{0,1\}\). In this model, each symbol (bit) sent by the transmitter experiences a distortion (flipping); that is, the received symbol (bit) can be flipped with a cross-over probability of \(p\in(0,1)\), but is otherwise received correctly. In contrast to the simplicity of the BSC, many information theoretical problems related to this model are still being investigated in the literature. For example, studying the behavior of the decoding error probabilities and characterization of them as a function of the codeword length \(n\), in the asymptotic for the entire region of coding rate \(R\), which requires knowing the analytic function of the so-called _channel reliability function (CRF)_[35], is still unknown.  
In addition, the error exponents for a binary symmetric channel in several settings are not yet completely characterized; cf. [35; 36] for further details. The K-identification problem considered in this work is the most generalized and difficult version of the identification problem [26]; therefore, it is rather evident that studying this topic for a general model may be exceedingly hard. However, we can obtain some insights into the effects of the size of goal messages, \(K\), by restricting our investigations to a basic/simple frame of model, i.e., the BSC. More specifically, such information is a theoretical endeavor dedicated to the basic BSC model, which can be useful in the subsequent aspects.  
**Upgrade to advanced models:** Often, studying an information theoretical problem begins with considering the most basic and simple abstract model. This allows the theorists to develop the required analytical tools and techniques in more straightforward manner and benefit the specific results as guides to the use and analysis of more advanced models. In other words, general/advanced models can often inherit/benefit analytical tools, techniques, and comprehensive steps that have been developed for the basic models. For example, studying the DI problem for a discrete memoryless channel (DMC) [32] was initiated/sparked by an earlier work in the literature for the BSC [33].  
**Error correction codes and modulation:** The simplicity of such a basic model with a binary alphabet often is favorable for an explicit code construction problem or for employing modulation techniques. This advantage facilitates the procedure of cultivating novel coding methods. For example, the widely used polar transmission codes are adopted initially for a binary input memoryless channel [31]. Therefore, the simplicity of the BSC model allows experts to utilize it as a promising candidate for evaluation/analyzing the performance of future error correction DKI codes.  
**Information theoretical characteristics:** Several advanced channel models for IoT applications can be simplified/specialized to a BSC. This allows information theorists to examine basic characteristics of such IoT systems (CRF, error exponent, critical rate, etc.) and acquire decent analytical insights needed for practical aspects such as modulation/detection design and explicit code construction. Therefore, studying the BSC effectively yields/suggest solutions for more advanced problems of IoT [37]. In addition, the BSC model is a useful model for studying network coding, which is an important technique in order to enhance the performance of a communication network [36]. Concrete modern scenarios in IoT systems that include the BSC model are telephone links, radio communication lines [37], implementation of _noise aggregation methods_ for physical layer security [4], _decision fusions_ for multi-route and multi-hop wireless sensor networks [38], and multi-hop networks [39].