In the following questions, wherever required, use the Galois fields $$\mathbb{F}_{2^{3}} = \{ 0, 1, \alpha, \alpha^{2}, \alpha^{3} = 1+\alpha, \alpha^{4} = \alpha+\alpha^{2}, \alpha^{5}=1+\alpha+\alpha^{2}, \alpha^{6}= 1+\alpha^{2} \} \, \text{and}$$ $$\begin{align*} \mathbb{F}_{2^{4}} = \{ & 0, 1, \alpha, \alpha^{2}, \alpha^{3}, \alpha^{4} = 1+\alpha, \alpha^{5} = \alpha+\alpha^{2}, \alpha^{6}=\alpha^{2}+\alpha^{3}, \alpha^{7}= 1+\alpha+\alpha^{3}, \\ & \alpha^{8} = 1+\alpha^{2}, \alpha^{9} =\alpha+\alpha^{3}, \alpha^{10}=1+\alpha+\alpha^{2}, \alpha^{11}=\alpha+\alpha^{2}+\alpha^{3},\\ & \alpha^{12}=1+\alpha+\alpha^{2}+\alpha^{3}, \alpha^{13}=1+\alpha^{2}+\alpha^{3}, \alpha^{14} = 1+\alpha^{3} \}. \end{align*}$$ 1. Let $\alpha$ be a primitive element of the Galois field $\mathbb{F}_{2^3}$. Find the minimal polynomial of $\alpha^{3}$

2. Find the generator polynomial $\mathbf{g}(X)$ for $(7,4)$ binary primitive one error-correcting code.

For questions 3 and 4 consider the same example which we discussed in the theory of this experiment (Example 3). For the $(15, 5, 7)$ triple-error-correcting BCH code, suppose that $\mathbf{v} = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]$ is transmitted, and the vector $\mathbf{r} = [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0]$ is received. Then $\mathbf{r}(X) = X^{3}+X^{5}+X^{12}$. Assume that we have performed Berlekamp's iterative algorithm up to the 3rd iteration (i.e, $\mu = 3$) and is provided in the Table 7. $\\$ 3. The minimum-degree polynomial $\sigma^{(4)}(X)$ determined at $\mu=4$th iteration for the above-mentioned event is

4. The value of the $\rho$ at $\mu=5 th$ iteration for the above-mentioned event is

5. The single error-correcting binary primitive BCH code $(n= 2^{m}-1)$ is

6. Consider the double-error correcting $(15, 7)$ BCH code. Assume that the codeword of all zeros $\mathbf{v} = [0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0]$ is transmitted and the received the vector $\mathbf{r} = [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]$. Find the error-location polynomial $\sigma(X)$, and values of $d_2, d_{3}$ by using Berlekamp's iterative algorithm.