Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Example Problem on Group Codes in Coding Theory
#### Problem 5: Show that the following code is a group code.
To solve this problem, you need to verify that the given code forms a group under binary addition. The elements of the code presented are:
\[
(00000), (00101), (10110), (10011)
\]
### Steps to Show it is a Group Code:
1. **Closure**: Verify that the addition (bitwise XOR) of any two elements results in another element within the code.
2. **Associativity**: Binary addition (bitwise XOR) is associative.
3. **Identity Element**: Confirm that there is an identity element (in this case, likely (00000)) within the set.
4. **Inverse**: Verify that each element has an inverse within the set such that element XOR inverse equals the identity element (00000).
When you work through these steps and confirm all criteria, you would then demonstrate that the code forms a group and is, hence, a group code.](https://content.bartleby.com/qna-images/question/fbb66fa7-7c22-4982-a22f-aaed542f65b3/f96ed350-51ed-4fa7-8e06-67eebedd2650/mkpm79me_processed.png)
Transcribed Image Text:### Example Problem on Group Codes in Coding Theory
#### Problem 5: Show that the following code is a group code.
To solve this problem, you need to verify that the given code forms a group under binary addition. The elements of the code presented are:
\[
(00000), (00101), (10110), (10011)
\]
### Steps to Show it is a Group Code:
1. **Closure**: Verify that the addition (bitwise XOR) of any two elements results in another element within the code.
2. **Associativity**: Binary addition (bitwise XOR) is associative.
3. **Identity Element**: Confirm that there is an identity element (in this case, likely (00000)) within the set.
4. **Inverse**: Verify that each element has an inverse within the set such that element XOR inverse equals the identity element (00000).
When you work through these steps and confirm all criteria, you would then demonstrate that the code forms a group and is, hence, a group code.
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