Understanding Generator and Parity Check Matrices

The generator matrix (G) and parity check matrix (H) are key components in linear block codes. They are linked by the fundamental equation:

$ GH^T = 0 \pmod{2} $

This equation means that any valid codeword, represented by a row vector in the code space generated by \( G \), must be orthogonal to the parity check vectors defined by the rows of \( H \). The transpose \( H^T \) is used in the formula.

For a linear block code denoted as (n, k):

• \( n \) is the length of each codeword.
• \( k \) is the length of the original message.
• \( G \) has dimensions \( k \times n \).
• \( H \) has dimensions \( (n-k) \times n \).

The question describes a (5, 1) repetition code, so \( n=5 \) and \( k=1 \). This means:

• \( G \) must be a \( 1 \times 5 \) matrix.
• \( H \) must be a \( (5-1) \times 5 = 4 \times 5 \) matrix.

Analyzing the Given Parity Check Matrix (H)

The parity check matrix \( H \) is provided as:

$ H = \begin{bmatrix} 1 & 0 & 0 & 0 & : 1 \\ 0 & 1 & 0 & 0 & : 1 \\ 0 & 0 & 1 & 0 & : 1 \\ 0 & 0 & 0 & 1 & : 1 \end{bmatrix} $

For calculation purposes, we consider the 5 columns. The rows relevant for the orthogonality check are:

• \( h_1 = [1, 0, 0, 0, 1] \)
• \( h_2 = [0, 1, 0, 0, 1] \)
• \( h_3 = [0, 0, 1, 0, 1] \)
• \( h_4 = [0, 0, 0, 1, 1] \)

The colons in the provided notation serve as visual separators.

Deriving the Generator Matrix (G)

Let the generator matrix \( G \) be \( G = [g_1, g_2, g_3, g_4, g_5] \). The condition \( GH^T = 0 \pmod{2} \) implies that \( G \) must be orthogonal to each row of \( H \). We check this orthogonality condition (dot product equals 0 mod 2) for each row \( h_i \):

Orthogonality Check with \( h_1 \)

$ G \cdot h_1 = [g_1, g_2, g_3, g_4, g_5] \cdot [1, 0, 0, 0, 1] = g_1(1) + g_2(0) + g_3(0) + g_4(0) + g_5(1) = 0 \pmod{2} $

$ g_1 + g_5 = 0 \pmod{2} $

This implies \( g_1 = g_5 \).

Orthogonality Check with \( h_2 \)

$ G \cdot h_2 = [g_1, g_2, g_3, g_4, g_5] \cdot [0, 1, 0, 0, 1] = g_1(0) + g_2(1) + g_3(0) + g_4(0) + g_5(1) = 0 \pmod{2} $

$ g_2 + g_5 = 0 \pmod{2} $

This implies \( g_2 = g_5 \).

Orthogonality Check with \( h_3 \)

$ G \cdot h_3 = [g_1, g_2, g_3, g_4, g_5] \cdot [0, 0, 1, 0, 1] = g_1(0) + g_2(0) + g_3(1) + g_4(0) + g_5(1) = 0 \pmod{2} $

$ g_3 + g_5 = 0 \pmod{2} $

This implies \( g_3 = g_5 \).

Orthogonality Check with \( h_4 \)

$ G \cdot h_4 = [g_1, g_2, g_3, g_4, g_5] \cdot [0, 0, 0, 1, 1] = g_1(0) + g_2(0) + g_3(0) + g_4(1) + g_5(1) = 0 \pmod{2} $

$ g_4 + g_5 = 0 \pmod{2} $

This implies \( g_4 = g_5 \).

Constructing the Generator Matrix G

From the orthogonality checks, we established that \( g_1 = g_5 \), \( g_2 = g_5 \), \( g_3 = g_5 \), and \( g_4 = g_5 \). Therefore, all elements of the generator vector must be equal: \( g_1 = g_2 = g_3 = g_4 = g_5 \).

For a (5, 1) code, \( k=1 \), meaning there is only one generator vector, and it must be non-zero. The simplest way to satisfy the condition \( g_1 = g_2 = g_3 = g_4 = g_5 \) is by choosing \( g_5 = 1 \).

This gives us:

• \( g_1 = 1 \)
• \( g_2 = 1 \)
• \( g_3 = 1 \)
• \( g_4 = 1 \)
• \( g_5 = 1 \)

So, the generator matrix is \( G = [1, 1, 1, 1, 1] \). This represents a (5, 1) repetition code, where the single message bit is repeated 5 times.

The resulting generator matrix is $G=[11111]$.