Understanding Minimum Hamming Distance

The minimum Hamming distance ($d_{min}$) of a linear block code is the smallest Hamming weight among all its non-zero codewords. For a linear code, this is equivalent to the minimum Hamming weight of the non-zero rows in its generator matrix G, or any non-zero linear combinations of these rows.

Generator Matrix G

The given generator matrix for the (6,3) code is:

$ G = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{bmatrix} $

Calculating Hamming Weights

We calculate the Hamming weight (number of 1s) for each row of the generator matrix G:

• Weight of row 1: $w(r_1) = w([1, 0, 0, 1, 0, 1]) = 3$
• Weight of row 2: $w(r_2) = w([0, 1, 0, 0, 1, 1]) = 3$
• Weight of row 3: $w(r_3) = w([0, 0, 1, 1, 1, 0]) = 3$

Linear Combinations of Rows

Next, we find the Hamming weights of the non-zero linear combinations of the rows (calculated modulo 2):

1. $r_1 + r_2$ = $[1, 0, 0, 1, 0, 1] \oplus [0, 1, 0, 0, 1, 1] = [1, 1, 0, 1, 1, 0]$. Weight = 4.

2. $r_1 + r_3$ = $[1, 0, 0, 1, 0, 1] \oplus [0, 0, 1, 1, 1, 0] = [1, 0, 1, 0, 1, 1]$. Weight = 4.

3. $r_2 + r_3$ = $[0, 1, 0, 0, 1, 1] \oplus [0, 0, 1, 1, 1, 0] = [0, 1, 1, 1, 0, 1]$. Weight = 4.

4. $r_1 + r_2 + r_3$ = $[1, 1, 0, 1, 1, 0] \oplus [0, 0, 1, 1, 1, 0] = [1, 1, 1, 0, 0, 1]$. Weight = 4.

Determining Minimum Hamming Distance ($d_{min}$)

The Hamming weights of all non-zero codewords generated (including rows of G and their combinations) are {3, 3, 3, 4, 4, 4, 4}.

The minimum value in this set is 3.

Therefore, the minimum Hamming distance $d_{min}$ is 3.