Let C1 be an n1 k linear systematic code with minimum distan

Let C_1 be an (n_1, k) linear systematic code with minimum distance d_1 and generator matrix G_1 = [P_1 I_k]. Let C_2 be an (n_2, k) linear systematic code with minimum distance d_2 and generator matrix G_2 = [P_2 I_k]. Consider an (n_1 + n_2, k) linear code with the following parity-check matrix: H = Show that this code has a minimum distance of at least d_1 + d_2.

Solution

The product code C is the code obtained by the following three-step encoding method. In n k the first step, k1 independent information bits are placed in each of k2 rows, thus creating a k2 × k1 rectangular array (see Figure 1a). In the second step, the k1 information bits in each of these k2 rows are encoded into a codeword of length n1 in C1, thus creating a 2 × n1 rectangular array (see Figure 1b). In the third step, the k2 information bits in each of the n1 columns are