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Consider the Hamming (7,4,3) linear block code with the parity bits:

P1=D1+D2+D4 ,
P2=D1+D3+D4 ,
P3=D2+D3+D4

Write the generator matrix G

User Hugohabel
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1 Answer

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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
]

User Robert Brisita
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