Cyclic codes

1. Let g(X)\mathbf{g}(X) is the generator polynomial of (n,k)(n,k) cyclic code. The codeword of the message polynomial u(X)\mathbf{u}(X) is v(X)=u(X)g(X)\mathbf{v}(X) = \mathbf{u}(X) \mathbf{g}(X). For the same message polynomial u(X)\mathbf{u}(X), the codeword obtained by shift register based encoder is v(X)=b(X)+Xnku(X)\mathbf{v}^{'}(X) = \mathbf{b}(X) + X^{n-k}\mathbf{u}(X). Here, b(X)\mathbf{b}(X) be the remainder obtained when the polynomial Xnku(X)X^{n-k}\mathbf{u}(X) is divided by g(X)\mathbf{g}(X). Is v(X)=v(X)\mathbf{v}(X) = \mathbf{v}^{'}(X) for all message polynomials?
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2. Which of the following codes is not a cyclic code?
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3. Consider (7,4)(7,4) cyclic code with the generator polynomial g(X)=1+X+X3\mathbf{g}(X) = 1+X+X^{3}. Using shift register based encoder, find the codeword of the message u=[0001]\mathbf{u} = \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}
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4. Consider the Meggitt decoder for a cyclic code given in Figure ef{fig:meggitt_decoder_quiz}. Suppose the content in the buffer register is [1111110]\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 0 \end{bmatrix} and in the buffer register is [101]\begin{bmatrix} 1 & 0 & 1 \end{bmatrix}. What will be the contents of the, buffer register, syndrome register and out put of the decoder at the next time instant?
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5. Consider (7,4)(7,4) cyclic code with the generator polynomial g(X)=1+X2+X3\mathbf{g}(X) = 1+X^{2}+X^{3}. The syndrome polynomial of r(X)=X6\mathbf{r}(X) = X^{6} is
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6. Consider (7,4)(7,4) cyclic code with the generator polynomial g(X)=1+X2+X3\mathbf{g}(X) = 1+X^{2}+X^{3}. For this code, the error pattern detection circuit (detector) of the decoder has to be designed in such a way that the output of the detector is 11 only if content of the syndrome register is
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