cryptography

Let C_1 be a linear [n, k_1, d_1] code and C_2 be a linear [n, k_2, d_2] code (of the same length) both over F = GF(q). Define the code C by C = {(c_1 | c_1 + c_2): c_1 belongs to C_1, c_2 belongs to C_2}. Let G_1 and G_2 be the generator matrices of C_1 and C_2, respectively. Prove that C is a linear code generated by [G_1 0 G_1 G_2]. Let H_1 and H_2 be parity-check matrices of C_1 and C_2, respectively. Show that [H_1 H_2 0 -H_2]. Show that C is a linear [2n, k, d]code where k = k_1 + k_2 and d = min{2d_1, d_2}. Let C_1 be a linear [n, k_1, d_1] code and C_2 be a linear [n, k_2, d_2] code (of the same length) both over F = GF(q). Define the code C by C = {(c_1 | c_1 + c_2): c_1 belongs to C_1, c_2 belongs to C_2}. Let G_1 and G_2 be the generator matrices of C_1 and C_2, respectively. Prove that C is a linear code generated by [G_1 0 G_1 G_2]. Let H_1 and H_2 be parity-check matrices of C_1 and C_2, respectively. Show that [H_1 H_2 0 -H_2]. Show that C is a linear [2n, k, d]code where k = k_1 + k_2 and d = min{2d_1, d_2}