Question

Question 12: If is a generator polynomial of a BCH code in $\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^n-1\right)$}\right.$, prove that divides $\left(x^n-1\right)$. [Hint: Exercises 11 and 8(b).]

Short answer

Answer

It has been proved that g(x) divides$x^n-1$ .

Step-by-step solution

Consider the results of 

Exercise 11: Let $K=Z_2\left(\alpha \right)$be a finite field of order $2^r$, whose multiplicative group is generated by . For each i, let$m_i\left(x\right)$ be the polynomial of over . If ,$\mathbf{n}\mathbf{=}{\mathbf{2}}^{\mathbf{r}}\mathbf{-}\mathbf{1}$ then each$m_i\left(x\right)$ divides .

Exercise 8: If${\mathbf{m}}_{\mathbf{1}}\left(\mathbf{x}\right)\mathbf{,}{\mathbf{m}}_{\mathbf{2}}\left(\mathbf{x}\right)\mathbf{,}...\mathbf{,}{\mathbf{m}}_{\mathbf{k}}\left(\mathbf{x}\right)$ are distinct monic irreducible in$F\left[x\right]$ such that each$m_i\left(x\right)$ divides $f\left(x\right)$, then $\mathrm{g}\left(\mathrm{x}\right):{\mathrm{m}}_1\left(\mathrm{x}\right),{\mathrm{m}}_2\left(\mathrm{x}\right),...,{\mathrm{m}}_{\mathrm{k}}\left(\mathrm{x}\right)$divides $f\left(x\right)$

Write the given data

Given that g(x)is a generator polynomial of a BCH code in $\raisebox{1ex}{${\mathrm{\mathbb{Z}}}_2\left[x\right]$}\!\left/ \!\raisebox{-1ex}{$\left(x^n-1\right)$}\right.$.

Determine the Proof

Here,$g\left(x\right)$ is a product of distinct minimal polynomials$m_i\left(x\right)$ .

According to Exercise 11, $m_i\left(x\right)$divides $x_n-1$

This implies$g\left(x\right)$ divides $x_n-1$.

Hence, the result.