### Contributions  
In this paper, we address identification systems whose encoders are deterministic and their receiver is required to conduct the K-identification job, i.e., spotting an object-t/event/message within a set of goal objects/events/messages with size \(K=2^{m}\kappa\) for some \(\kappa\in[0,1)\). We assume that the communication over \(n\) channel uses are independent of each other, and the noise is additive Bernoulli process. We formulate the problem of DKI over the BSC with and without Hamming weight input constraint. Our primary goal is to study the BSC's DKI capacity region. This study specifically contributes the subsequent contributions:  
\(\diamondsuit\)Generalized identification modelWe examine the BSC, in which the size of the goal message set, \(K\), may scale with the codeword length, \(n\). As a consequence, this model incorporates the DI with \(K=1\), and DKI with constant \(K>1\). Therefore, we can confirm whether asymptotic codeword lengths allow for reliable identification, even when the goal message set grows in size, using our suggested generalized model. As far as is known by the authors' knowledge, no previous research has been conducted on a generalized DKI model in the literature. \(\diamondsuit\)Codebook scaleWe prove that, for K-identification over the BSC with deterministic encoding, the codebook size grows in \(n\), similarly to that of the DI problem (\(K=2^{0}=1\)) [32; 33] and the TR problem [17] over the same channel, namely _exponentially_ in the codeword length \(n\), i.e., \(\sim\!\!2^{nR}\), where \(R\) is the DKI coding rate, even when the size of the goal message set grows _exponentially_ in \(n\), i.e., \(K=2^{n\kappa}\), where \(\kappa\in[0,1)\) is the identification goal rate, and certain functions of the channel statistics and input restrictions set upper bounds on it. This result implies that one can extend the collection of goal messages for identification without compromising the codebook's scalability. * Capacity formula: We derive inner and outer bounds on the DKI capacity region for constant \(K\geq 1\) and growing \(K=2^{n\kappa}\), for the BSC with and without Hamming weight constraints. Our capacity bounds _reflect_ the impact of the channels statistics, i.e., cross-over probability and the input constraint \(A\) in the _optimal_ scale of the codebook size, i.e., \(2^{nR}\). In particular, in the coding procedure, we define a parameter \(\beta\in(0,1)\), referred to as the distinction property of the codebook which adjust the Hamming distance property for the constructed codebook. Then, assuming a given codebook distinction, \(\beta\), a channel with asymptotic small cross-over probability (i.e., an almost perfect channel) causes the feasible range for the goal identification rate \(\kappa\) to shrink; that is, the capability of the BSC for K-identification decreases, which is unfavorable. On the other hand, when the cross-over probability increases and converges to its maximum possible values, i.e., \(\varepsilon\to 1/2\) (almost pure noisy channel), then the feasible range for \(\kappa\) begins to enlarge favorably. This observation can be interpreted as follows: The channel noise can be _exploited_ as an additional inherent source embedded in the communication setting for performing the K-identification task with a larger value of \(K\). This observation is in contrast to previous results for DKI over the slow fading channel [51], or the DI for Gaussian and Poisson channels [32; 48; 52], where capacity bounds were shown to be independent of the input constraints or the channel parameters. We demonstrate that the suggested upper and lower bounds on attainable rates \((R,\kappa)\) are independent of \(K\) for constant \(K\), whereas they are functions of the goal identification rate \(\kappa\) for increasing goal message sets.