Question

Show that the standard generator matrix

G= $\mathbf{\left(}\begin{array}{l}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\mathbf{1}& \mathbf{0}\end{array}& \mathbf{1}& \mathbf{0}\end{array}& \mathbf{1}& \mathbf{0}& \mathbf{1}\end{array}& \mathbf{0}& \mathbf{1}& \mathbf{0}\end{array}\\ \begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\mathbf{0}& \mathbf{1}\end{array}& \mathbf{0}\end{array}& \mathbf{1}\end{array}& \mathbf{0}& \mathbf{1}& \mathbf{0}\end{array}& \mathbf{1}& \mathbf{0}& \mathbf{1}\end{array}\end{array}\mathbf{\right)}$

generates the $\mathbf{(}\mathbf{10}\mathbf{,}\mathbf{2}\mathbf{)}$repetition code in Example$\mathbf{5}$ . [Hint: See the hint for Exercise$\text{7}$ .]

Short answer

The standard generator matrix

$G=\left(\begin{array}{l}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}1& 0\end{array}& 1& 0\end{array}& 1& 0& 1\end{array}& 0& 1& 0\end{array}\\ \begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}0& 1\end{array}& 0\end{array}& 1\end{array}& 0& 1& 0\end{array}& 1& 0& 1\end{array}\end{array}\right)$

generates the $\text{(10,2)}$ repetition code.

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.

The Hamming weight of an element of B(n) in the group B(n) is the number of non-zero coordinates in that element.

Identify the repetitive code

Consider the standard generator matrix as follows:

$\text{G=}\left(\begin{array}{l}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{1}& \text{0}\end{array}& \text{1}& \text{0}\end{array}& \text{1}& \text{0}& \text{1}\end{array}& \text{0}& \text{1}& \text{0}\end{array}\\ \begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\text{0}& \text{1}\end{array}& \text{0}\end{array}& \text{1}\end{array}& \text{0}& \text{1}& \text{0}\end{array}& \text{1}& \text{0}& \text{1}\end{array}\end{array}\right)$

The following are the elements in the$\text{(10,2)}$codes:

$\text{{}\left(\begin{array}{cc}\text{0}& \text{0}\end{array}\right)\text{,}\left(\begin{array}{cc}\text{0}& \text{1}\end{array}\right)\text{,}\left(\begin{array}{cc}\text{1}& \text{0}\end{array}\right)\text{,}\left(\begin{array}{cc}\text{1}& \text{1}\end{array}\right)\text{}}$

As a result, the following are the related code words:

$\begin{array}{c}\left(\begin{array}{cc}\text{0}& \text{0}\end{array}\right)\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}\right)\text{=}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{0}& \text{0}\end{array}& \text{0}& \text{0}\end{array}& \text{0}& \text{0}& \text{0}\end{array}& \text{0}& \text{0}& \text{0}\end{array}\right)\\ \left(\begin{array}{cc}\text{0}& \text{1}\end{array}\right)\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}\right)\text{=}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{0}& \text{1}\end{array}& \text{0}& \text{1}\end{array}& \text{0}& \text{1}& \text{0}\end{array}& \text{1}& \text{0}& \text{1}\end{array}\right)\\ \left(\begin{array}{cc}\text{1}& \text{0}\end{array}\right)\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}\right)\text{=}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{1}& \text{0}\end{array}& \text{1}& \text{0}\end{array}& \text{1}& \text{0}& \text{1}\end{array}& \text{0}& \text{1}& \text{0}\end{array}\right)\\ \left(\begin{array}{cc}\text{1}& \text{1}\end{array}\right)\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{cc}\begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}& \begin{array}{l}\text{1}\\ \text{0}\end{array}\end{array}& \begin{array}{l}\text{0}\\ \text{1}\end{array}\end{array}\right)\text{=}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{1}& \text{1}\end{array}& \text{1}& \text{1}\end{array}& \text{1}& \text{1}& \text{1}\end{array}& \text{1}& \text{1}& \text{1}\end{array}\right)\end{array}$

Thus, the given standard generator matrix produces the$\text{(10,2)}$ repetitive code

$\begin{array}{c}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{0}& \text{0}\end{array}& \text{0}& \text{0}\end{array}& \text{0}& \text{0}& \text{0}\end{array}& \text{0}& \text{0}& \text{0}\end{array}\right)\text{,}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{0}& \text{1}\end{array}& \text{0}& \text{1}\end{array}& \text{0}& \text{1}& \text{0}\end{array}& \text{1}& \text{0}& \text{1}\end{array}\right)\\ \left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{1}& \text{0}\end{array}& \text{1}& \text{0}\end{array}& \text{1}& \text{0}& \text{1}\end{array}& \text{0}& \text{1}& \text{0}\end{array}\right)\text{,}\left(\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\text{1}& \text{1}\end{array}& \text{1}& \text{1}\end{array}& \text{1}& \text{1}& \text{1}\end{array}& \text{1}& \text{1}& \text{1}\end{array}\right)\end{array}$

Therefore, the result stated in the question is proved.