## 5 Future Directions and Summary  
In this work, the deterministic K-identification problem for IoT systems was studied. The results obtained in this paper can serve as a model for tasks that are based on an event recognition within the context of future IoT applications. Specifically, we consider IoT systems that can be modeled by the binary symmetric channel. For this setup, we established inner and outer bounds on the DKI capacity region with/without the Hamming weight constraint for a codebook size of \(M(n,R)=2^{nR}\). Our results in this work regarding the DKI capacity for the BSC model unveiled that the conventional exponential scale of \(2^{nR}\) considered for the DI [32] and TR problems [17], is the appropriate scale for the codebook size of the DKI problem of the BSC with/without Hamming weight constraint. This observation is was proved by finding a suitable ball covering for an \(n\)-dimensional Hamming hyper ball or the entire input set in the same line of arguments as that for the basic _Gilbert bound_ method. In particular, in the presence of a Hamming weight constraint \(A\), we pack hyper balls with radius \(\left\lfloor n\beta\right\rfloor\), for some \(\beta\in(0,1)\) inside a larger Hamming hyper ball, which results in \(\sim\)\(2^{nH(A)}\) codewords. We remind you that the scale of the codebook for DKI over the BSC is lower than that for the DKI over slow fading channels [51] or the DI over Poisson channel with and without ISI [48; 52]. Moreover, we find out that the BSC features an _exponentially_ large set of the goal messages set, in the codeword length, \(n\), i.e., \(2^{nR}\); and characterize the entire feasible range on the goal identification rate \(\kappa\) as a function of the channels statistic \(\varepsilon\) and the Hamming constraint (for \(0<A<1/2\)). For the converse part, a similar approach as our previous work for DI over the DMC [32] is followed. That is, for the case where a non-trivial Hamming weight constraint is present (\(0<A<1\)), we establish an one-to-one mapping between the message set and the feasible set induced by the Hamming weight constraint. In particular, we exploit the method of _proof by the contradiction_. Namely, we first assume that two generic different messages \(i_{1}\) and \(i_{2}\) share the common codewords, and then show that such an assumption leads to a contradiction regarding the sum of the error probabilities, i.e., we derive that the sum of the sort I and sort II error probabilities converges to one. Hence, the falsehood of the early assumption is guaranteed, and the total number of messages \(M=2^{nR}\) is bounded by the size of the feasible input set, i.e., \(M\leq 2^{nH(A)}\). For the case where \(A\geq 1\), (absent of a Hamming constraint), a similar line of argument can be applied in order to establish the one-to-one function. There are numerous ways to expand upon the findings we have showcased in this manuscript. Some of the possible topics for the future research are as follows:  
* Explicit code construction: In this paper, we mainly address the determination of basic performance constraints for the DKI for the BSC with/without Hamming weight constraint, where an explicit code construction was not investigated. That is, in the achievability proof, we only guarantee the existence of a code without suggesting a systematic method for construction of the code. Therefore, an important direction for research may be explicit construction of K-identification codes for the BSC and development of efficient encoding and low complexity decoding schemes. Furthermore, the efficiency of such concrete designs can be measured versus the information theoretical bounds derived in this paper. * Generalized channel models: We consider in this work one of the simplest and most basic channel model, namely the BSC in the absence of channel state, memory, or feedback. Therefore, our result can be extended to a DMC (with or without memory/feedback), compound, and arbitrary varying channels, which are _generalizations_ of the BSC.