Question

Question:- If $\mathbf{f}\left(\mathbf{x}\right)\in {\mathbf{Z}}_{\mathbf{2}}\left[\mathbf{x}\right]$and$\alpha$ is an element in some extension field of ,${\mathbf{Z}}_{\mathbf{2}}$ prove$\mathbf{k}⩾\mathbf{1}\mathbf{,}\mathbf{f}\left({\mathbf{\alpha }}^{\mathbf{2}\mathbf{k}}\right)\mathbf{=}\mathbf{f}{\left({\mathbf{\alpha }}^{\mathbf{k}}\right)}^{\mathbf{2}}$ [Hint Lemma 11.24 ]

Short answer

Answer: -

It is proved that: $\mathrm{k}⩾1,\mathrm{f}\left({\mathrm{\alpha }}^{2\mathrm{k}}\right)=\mathrm{f}{\left({\mathrm{\alpha }}^{\mathrm{k}}\right)}^2$ .

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.

Prove the given statement

Proving:

Assuming that$f\left(x\right)\in {\mathrm{\mathbb{Z}}}_2\left[x\right]$and$\alpha$is an element.

It is in some extension field of${\mathrm{\mathbb{Z}}}_2$.

Then the value ofis:

$f\left(x\right)=a_0+a_{1}x+a_2{x}^2+.....+a_n{x}^n$

For the value of$k⩾1$the equation is proved as:

$$\begin{gathered} f{\left(a^k\right)}^2={\left(a_0+a_1{x}^k+a_2{x}^{2k}+.....+a_n{x}^{nk}\right)}^2 \\ =a_0^2+a_1^2{x}^{2k}+a_2^2{x}^{4k}+.....+a_n^2{x}^{2nk} \\ =a_0^2+a_1{x}^{2k}+a_2{\left({\alpha }^{2k}\right)}^2+.....+a_n{\left({\alpha }^{2k}\right)}^n \\ =f\left({\alpha }^{2k}\right) \end{gathered}$$

Therefore, it is proved.