## 3 System Model and Preliminaries
### DKI Coding for the BSC  
The definition of a DKI code for the BSC, \(\mathcal{B}\), is given below. **Definition 1** ( Bsc Dki Code). : _An \((n,M(n,R),K(n,\kappa),e_{1},e_{2})\)-BSC-DKI code for a BSC \(\mathcal{B}\) for integers \(M(n,R)\) and \(K(n,\kappa)\), where \(n\) and \(R\) are the codeword length and coding rate, respectively, is defined as a system \((\mathcal{C},\mathcal{F}_{\text{K}})\), which consists of a codebook \(\mathcal{C}=\{\mathbf{c}_{i}\}_{i\in[\mathbb{M}]}\subseteq\mathbf{H}^{n}\), with \(\mathbf{c}_{i}=(c_{iL})|_{t=1}^{n}\subseteq\mathbf{H}^{n}\), such that \(n^{-1}\sum_{t=1}^{n}c_{iL}\leq A,\forall i\in[\mathbb{M}]\) and a decoder (We recall that the decoding  
Figure 1: Bit transition graph over a BSC. Each bit is flipped independently of other bits, with a cross-over probability of \(\varepsilon\in(0,1/2)\). sets for the DKI problem, similarly to that for the RI problem, may have in general intersection; however, to guarantee a vanishing sort II error probability for the asymptotic codeword lengths \(n\), an optimal decoder may be defined in a way such that the size of such intersection regions becomes negligible) \(\mathcal{F}_{\mathsf{K}}\subseteq\mathbf{H}^{n}\), where \(\mathsf{K}\) is an arbitrary subset (recall that the system (family) of all subsets of the set \(\llbracket M\rrbracket\), of size \(K\), is \(\{\mathsf{K}\subseteq\llbracket M\rrbracket\;;|\mathsf{K}|=K\}\); note that \(|\{\mathsf{K}\subseteq\llbracket M\rrbracket\;;|\mathsf{K}|=K\}|=\binom{M}{K}\), and the error requirements, required by the DKI code definition, apply to every possible choice of the set \(\mathsf{K}\) with \(K\) arbitrary messages among all \(\binom{M}{K}\) cases) of \(\llbracket M\rrbracket\) with size \(K\), see Figures 2 and 3._  
_The encoder sends codeword \(\mathbf{c}_{i}\), given a message \(i\in\llbracket M\rrbracket\), and the decoder's job is to solve a binary hypothesis: was \(j\in\mathsf{K}\) a goal message that was sent or not? See Figure 3. There exist two sorts of errors that may happen:_  
* _Sort I Error Event: Rejection of the actual message;_ \(i\in\mathsf{K}\)_._
* _Sort II Error Event: Acceptance of a wrong message;_ \(i\
otin\mathsf{K}\)_._
* _The associated error probabilities of the DKI code_ \((\mathcal{C},\mathcal{F})\) _read_ \[P_{e,1}(i,\mathsf{K})=\Pr(\mathbf{Y}\in\mathcal{F}_{\mathsf{K}}^{c}\,|\,\mathbf{ x}=\mathbf{c}_{i}) =1-\sum_{\mathbf{y}\in\mathcal{F}_{\mathsf{K}}}W^{n}(\mathbf{y}\,|\,\mathbf{c}_{ i}),\quad\forall i\in\mathsf{K}\ \ \mbox{miss-identification},\] (8) \[P_{e,2}(i,\mathsf{K})=\Pr(\mathbf{Y}\in\mathcal{F}_{\mathsf{K}}\,|\, \mathbf{x}=\mathbf{c}_{i})=\sum_{\mathbf{y}\in\mathcal{F}_{\mathsf{K}}}W^{n} (\mathbf{y}\,|\,\mathbf{c}_{i}),\quad\forall i\
otin\mathsf{K}\ \ \mbox{false identification},\] (9)  
Figure 3: A DKI configuration with \(K=4\) and a goal message set \(\mathsf{K}=\{2,4,5,7\}\) is displayed. The channel’s output is located in the union of each individual decoder \(\mathsf{T}_{j}\) (marked in blue) in the correct identification event, where \(j\) is a member of the goal message set. If the channel output is seen in the complement of the union of distinct decoders that the codeword’s index belongs to, a sort I error event takes place. When the transmitted codeword’s index does not _not_ belong to \(\mathsf{K}\), and the channel output is recognized in the union of the individual decoders \(\mathsf{T}_{j}\), with \(j\in\mathsf{K}\), an error event of sort II occurs. Figure 2: System model for DKI communication setting over a BSC. Employing a deterministic encoder at the transmitter, the message \(i\) is mapped to the codeword \(\mathbf{c}_{i}=(c_{i,j})|_{i=1}^{n}\) using a deterministic function. The decoder at the receiver is provided with an arbitrary goal message set \(\mathsf{K}\), and given the channel output \(\mathbf{Y}=(Y_{t})|_{t=1}^{n}\), it asks whether or not \(i\) belongs to \(\mathsf{K}\).