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12E

Expert-verifiedFound in: Page 471

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**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).]**

**Answer**

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

__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 $f\left(x\right)$

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)$}\right.$.

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.

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