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

See similar textbooks

Related questions

Q: 810 and 1410, what will be their minimum binary length to avoid overflow when calculating A-B using…

A: Lets see the solution.

Q: Q2. Explain step by step the process of hamming code detection and correction by using data as…

A: According to the information given;- We have to define the process of hamming code detection and…

Q: Calculate (A – B), and (-A – B) for the following pair of binary numbers, assuming a two's…

A: 4. Given: 01111010 00000111 Find: A + B = 01111010 + 00000111 = 10000001 A - B: Take 2's…

Q: Consider two binary bit vectors x and y, each of length of n bits. What is the maximum and minimum…

A: Jaccard Coefficient:- A statistic used to determine how similar two groups are is the Jaccard…

Q: If we use the 128-bit binary field extension F 128 to decode the 127-bit binary BCH code at a…

A: The 127-bit binary BCH code can be decoded using the 128-bit binary field extension F 128, which…

Q: Given the above table: Assuming unsigned 4-bit representation of numbers, under what conditions…

A: Given that, The table contains two values A and B and their sum in both binary and decimal form. The…

Q: In this problem the generator G (=1001). Given G answer the following questions: a) Why can it…

A: a)

Q: Given a single-precision IEEE 754 representation of a floating point number as: 0x40700000, do the…

A: IEEE 754 is a standard for specifying the arithmetic behavior of floating-point integers and…

Q: In defining the Hamming distance for a code, we choose to usethe minimum (Hamming) distance between…

A: There is always a risk of losing is critical during the transition from one system to another…

Q: Consider the binary BCH code with length 127 and expected distance 6 using the binary field…

A: Introduction: The BCH codes, abbreviated for Bose–Chaudhuri–Hocquenghem codes, are cyclic…

Q: Two's complement (2'sC) code

A: Fixed-point code represents numbers with a fixed number of bits dedicated to the integer and…

Q: If we use the 128-bit binary field extension F 128 to decode the 127-bit binary BCH code at a…

A: The 127-bit binary BCH code with a distance of 6 can correct up to 2 errors. If we use the 128-bit…

Q: B={1,2,3,4,5,6,7,8,9,10} Using the n-bit binaries to which subset in the power set of the ordered…

A: The bit string 1100000011 represents the subset {1,2,9,10} in the power set of the ordered set B.…

Question

### 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

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Part a

VIEW

Step 2: Part b, c and d

VIEW

Solution

VIEW

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.

Similar questions

H - For the IEEE 754 single-precision floating point, write the hexadecimal representation for the following decimal values: (i)–1.0 (ii)– 0.0 (iii)256.015625
Q2. Find out the bit pattern of hamming code considering data bits="11011011". Also explain the phenomenon of error correction considering the 5rd bit (from the left) got inverted during transmission.
For an 8 bit string 01011011, determine the Hamming bits required and show that if there is an error in the last bit it received, it can be corrected using Hamming codes.
Suppose the following Hamming Codes were created using even parity, determine if the code is correct or not. If not correct, find the correct Hamming Code. c.) 010101100011
Consider the binary BCH code of length 127 with intended distance 6 utilizing the binary field extension F 128 of size 128. What is the code's size, and can you guarantee a minimum distance?
Given the Djistrka Matrix below, construct the corresponding digraph and explain the exact properties it exhibit. 0 100 110 1 1 10 0110101000 1 0101011 1100 00 10101000 1 01101110100 0 1001000110
Calculate 2^1023 and 2^(-1024) for IEEE double precision, the maximum for overflow and also with 52 bits mantissa, the number of digits for double precision.
Convert (158)10 to binary, using long division method, then convert your binary solution to both octal and hexadecimal using the bit grouping shortcut.
Consider the binary BCH code of length 127 with intended distance 6 utilizing the binary field extension F 128 of size 128. What is the code's size, and can you guarantee a minimum distance?