Consider the (6,2) code generated by the matrix 11010 01101 G = -11 (a) Calculate and list all of the codewords c = mG for this code, together with the Ham- ming weight for each codeword by filling in a table in the form given below. wH (c) m с (b) What is the minimum distance for this code? How many bit errors can this code cor- rect? (c) Tabulate that part of the standard array to correct all single bit errors for this code. Indicate the value of the syndrome for each row. r 110111 110010 101001 000001 (d) Assume the following values of r have been received. Calculate the corresponding c and m for each of the values of r below. с m
Consider the (6,2) code generated by the matrix 11010 01101 G = -11 (a) Calculate and list all of the codewords c = mG for this code, together with the Ham- ming weight for each codeword by filling in a table in the form given below. wH (c) m с (b) What is the minimum distance for this code? How many bit errors can this code cor- rect? (c) Tabulate that part of the standard array to correct all single bit errors for this code. Indicate the value of the syndrome for each row. r 110111 110010 101001 000001 (d) Assume the following values of r have been received. Calculate the corresponding c and m for each of the values of r below. с m
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![### Understanding Linear Block Codes
**Consider the (6, 2) code generated by the matrix**
\[ G = \begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 1 & 1 & 0 & 1
\end{bmatrix} \]
**Steps to Compute Various Parameters and Correct Errors:**
**(a) Calculate and list all of the codewords \( c = mG \) for this code, together with the Hamming weight for each codeword by filling in a table in the form given below.**
| m | c | \( w_H(c) \) |
|-------|-------------|--------------|
| | | |
**(b) What is the minimum distance for this code? How many bit errors can this code correct?**
**(c) Tabulate that part of the standard array to correct all single bit errors for this code. Indicate the value of the syndrome for each row.**
**(d) Assume the following values of \( r \) have been received. Calculate the corresponding \( c \) and \( m \) for each of the values of \( r \) below.**
| r | c | m |
|-------------|-------------|-----|
| 110111 | | |
| 110010 | | |
| 101001 | | |
| 000001 | | |
**Explanation:**
1. **Code Matrix (G):** This matrix is used to generate the codewords. Each codeword is a linear combination of the rows of matrix \( G \).
2. **Part (a):** You must calculate the codewords by multiplying each possible message vector \( m \) by the generator matrix \( G \) and then count the Hamming weight \( w_H(c) \) for that codeword.
3. **Part (b):** The minimum distance of the code \( d_{min} \) is the smallest Hamming distance between any pair of distinct codewords. The code can correct up to \( \left\lfloor \frac{d_{min} - 1}{2} \right\rfloor \) bit errors.
4. **Part (c):** A standard array is a way to systematically list](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aaf72ae-8073-4cb4-ae75-7d87fdc3f506%2F45aee0d4-b1dd-4d93-adbf-1a2667a18cf3%2Fwcaj4j4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Linear Block Codes
**Consider the (6, 2) code generated by the matrix**
\[ G = \begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 1 & 1 & 0 & 1
\end{bmatrix} \]
**Steps to Compute Various Parameters and Correct Errors:**
**(a) Calculate and list all of the codewords \( c = mG \) for this code, together with the Hamming weight for each codeword by filling in a table in the form given below.**
| m | c | \( w_H(c) \) |
|-------|-------------|--------------|
| | | |
**(b) What is the minimum distance for this code? How many bit errors can this code correct?**
**(c) Tabulate that part of the standard array to correct all single bit errors for this code. Indicate the value of the syndrome for each row.**
**(d) Assume the following values of \( r \) have been received. Calculate the corresponding \( c \) and \( m \) for each of the values of \( r \) below.**
| r | c | m |
|-------------|-------------|-----|
| 110111 | | |
| 110010 | | |
| 101001 | | |
| 000001 | | |
**Explanation:**
1. **Code Matrix (G):** This matrix is used to generate the codewords. Each codeword is a linear combination of the rows of matrix \( G \).
2. **Part (a):** You must calculate the codewords by multiplying each possible message vector \( m \) by the generator matrix \( G \) and then count the Hamming weight \( w_H(c) \) for that codeword.
3. **Part (b):** The minimum distance of the code \( d_{min} \) is the smallest Hamming distance between any pair of distinct codewords. The code can correct up to \( \left\lfloor \frac{d_{min} - 1}{2} \right\rfloor \) bit errors.
4. **Part (c):** A standard array is a way to systematically list
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