Question

Question: Show that the generator polynomial for the BCH code with $\mathit{t}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}\mathbf{,}\mathit{r}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{,}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{15}$is$\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{+}\mathit{x}\mathbf{+}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{5}}\mathbf{+}{\mathit{x}}^{\mathbf{8}}\mathbf{+}{\mathit{x}}^{\mathbf{10}}$ [Hint: Exercises 3-5 may be helpful.]

Short answer

Answer:

The polynomial is $g\left(x\right)=1+x+x^2+x^4+x^5+x^8+x^{10}$

Step-by-step solution

Conceptual Introduction

Coding theory is the study of a code's characteristics and how well-suited it is to various applications. Data compression, cryptography, error detection, and correction, as well as data transport and storage, all depend on codes.

Find the generator polynomial.

Let the values be:

$m_1\left(x\right),m_2\left(x\right),......,m_6\left(x\right)\in Z_2\left[x\right]$

To calculate we find minimum polynomial of $\alpha ,{\alpha }^2,{\alpha }^3,{\alpha }^4$over z₂

$m_1\left(x\right)=1+x+x^2$

Find minimum polynomial in α3 

We write minimal polynomial as:

$m_3\left(x\right)=1+x^2+x^3+x^6+x^8$

Hence, the polynomial is:

$$\begin{gathered} g\left(x\right)=m_1\left(x\right)·m_3\left(x\right) \\ =\left(1+x+x^2\right)\left(1+x^2+x^3+x^6+x^8\right) \\ =1+x+x^2+x^4+x^5+x^8+x^{10} \end{gathered}$$

Thus, the polynomial is: $g\left(x\right)=1+x+x^2+x^4+x^5+x^8+x^{10}$.