## 3 Proposed Solution
### Channel Modeling  
In this paper, the noisy channel is simulated with Gilbert-Elliot (GE) channel, which refers to a wide class of finite-state fading channels that model communication links with memory. This model is extensively simulated in research studies to find packet losses [52], frame losses [53], and bit losses [54], hence making it an appropriate choice for modeling the impact of errors within a noisy wireless channel [55]. The GE model is based on a discrete-time hidden Markov chain with two states that are the Good or Gap (\(G\)) state and the Bad or Burst (\(B\)) state. Let \(S=\{G,B\}\) be the state space of the transmission channel. The probabilities of error occurrence in \(G\) and \(B\) states are considered as \(P(G)\) and \(P(B)\), respectively [56]. The GE model is computationally expensive when considering error events in both states. For the sake of simplicity and efficiency, we have implemented a fixed GE model by assuming the probability of errors in the \(G\) state is fixed. Accordingly, \(P(G)=0\) represents that the event (error occurrence) in the \(G\) state will not happen, and all bits will be transferred correctly in the \(G\) state, while \(P(B)=1\) represents that the event (error occurrence) in a \(B\) state will happen. For inducing a small number of error bits, we assume \(P(G)>P(B)\) that the \(G\) state has a higher probability compared to the probability of error occurrence in the \(B\) state. Along with the probability of error occurrence, the transition probability (probability of switching from one state to another state) is also considered, which are of two types, i.e., _self-transition_ and _cross-transition probability_. In _self-transition probabilities_, \(P_{GG}\) is the probability of a transition from \(G\) to \(G\) state again, while the probability of a transition from \(B\) to \(B\) state again is \(P_{BB}\). In the _cross-transition probabilities_, \(P_{GB}\) denotes the probability of some \(G\) state next entering the \(B\) state and \(P_{BG}\) denotes the probability of some \(B\) state next entering the \(G\) state [57]. For example, when a current bit is received with an error (\(B\)) and the previous bit was transmitted correctly (\(G\)), the transition from \(G\) state to \(B\) state takes place and vice versa for \(B\) state to \(G\) state. The sum of all the probabilities for a particular state is always \(1\), such as \(P_{BB}+P_{BG}=1\) and \(P_{GG}+P_{GB}=1\). A state transition diagram of the GE channel model is given in Figure 2.  
Figure 1: Bit permutation round: (a) actual bit positions, (b) bits are right-shifted circularly by three bit positions. Figure 2: State Transitions of Gilbert–Elliot Channel Model. The channel states will evolve according to a Markov chain. The self-transition probabilities (3) and cross-transition probabilities (4) are computed as follows:  
\[P_{BB}=1-P_{BG}\qquad\text{ and }\qquad P_{GG}=1-P_{GB} \tag{3}\] \[P_{BG}=1-P_{BB}\qquad\text{ and }\qquad P_{GB}=1-P_{GG} \tag{4}\]  
Moreover, GE Model uses a \(2\times 2\) transition matrix of two states \(G\) and \(B\), at time \(T\), to determine the state transition probabilities. \[M=\left[ \begin{array}{cc}P_{GG}&P_{GB}\\ P_{BG}&P_{BB}\end{array} \right] \tag{5}\]  
\[M=\left[ \begin{array}{cc}1-P_{GB}&P_{GB}\\ P_{BG}&1-P_{BG}\end{array} \right] \tag{6}\]  
Furthermore, the stationary probabilities for the \(G\) and \(B\) states are denoted by \(\pi_{G}\) and \(\pi_{B}\) respectively. The stationary distribution of a Markov chain is a probability distribution that remains unchanged in the Markov chain as time progresses. Typically, it is represented as a row vector \(\pi\) whose entries are probabilities summing to 1 as \(\pi_{G}+\pi_{B}=1\), and transition matrix \(\Pi=M_{X}^{\top}\times\Pi\)[58]. Now, \(\pi_{G}=P_{GG}\times\pi_{G}+P_{BG}\times\pi_{B}\) and \(\pi_{B}=P_{GB}\times\pi_{G}+P_{BB}\times\pi_{B}\), so finally, the stationary probabilities for the G and B state are given in (7).