Question

Show that 1 X 4 standard generator matrix$\left(1111\right)$generates the code in Example 1.

Short answer

The $1\times 4$ standard generator matrix $\left(\text{1111}\right)$generates the code $\left(\text{0000}\right)$and $\left(\text{1111}\right)$.

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.

Code generation

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

Considering, the$\text{1×4}$standard generator matrix as follows:$\left(\text{1111}\right)$

Then the elements in the as follows:$\left(41\right)$codes are as follows:$\{0,1\}$

Thus, the corresponding code words are given as follows:

(0)$\left(\text{1111}\right)\text{=(0000)}$

(1)$\left(\text{1111}\right)\text{=}\left(\text{1111}\right)$

Thus, the given standard generator matrix produces code$\left(\text{0000}\right)$and$\left(\text{1111}\right)$

Therefore, the required result in the question is shown.