Final answer:
The generator matrix for a Hamming (7,4,3) code is a 7x4 matrix where the identity matrix represents data bits and the parity matrix is constructed from the parity bit equations, resulting in the matrix G listed above.
Step-by-step explanation:
The question deals with generating the generator matrix G for a Hamming (7,4,3) code which is a type of linear block code used in error detection and correction. The parity bits P1, P2, and P3 are defined as: P1=D1+D2+D4, P2=D1+D3+D4, and P3=D2+D3+D4. For a Hamming (7,4,3) code, the generator matrix G is constructed using the data bits and parity bits equations provided.
The generator matrix G for this code can be written in the form of a 7x4 matrix where the left side is a 4x4 identity matrix representing the data bits, and the right side is a 3x4 matrix representing the parity check equations:
G = [ I | P ]
Here, I is the identity matrix, and P is the matrix constructed from the parity bit equations. The generator matrix G in its full form is:
G = [
1 0 0 0 | P1
0 1 0 0 | P2
0 0 1 0 | P3
0 0 0 1 | P4
]
From the parity bit equations given, we can derive:
- P1 is represented by the column vector (1 1 0 1)
- P2 is represented by the column vector (1 0 1 1)
- P3 is represented by the column vector (0 1 1 1)
So, the full generator matrix G will be:
G = [
1 0 0 0 | 1 1 0 1
0 1 0 0 | 1 0 1 1
0 0 1 0 | 0 1 1 1
0 0 0 1 | 1 1 1 0
]
Which corresponds to:
G = [
1 0 0 0 1 1 0 1
0 1 0 0 1 0 1 1
0 0 1 0 0 1 1 1
0 0 0 1 1 1 1 0
]