If the code generator polynomial is g(x) = 1 + x^2 for a (5,3) code, find the linear block code generator matrix G.

To find the linear block code generator matrix G for a (5,3) code with the given code generator polynomial g(x) = 1 + x^2, we can follow these steps:

Step 1: Determine the number of rows and columns in the generator matrix. In this case, since it's a (5,3) code, the number of rows will be equal to the block length (5), and the number of columns will be equal to the number of information bits (3).

Step 2: Write down the powers of the variable x up to the degree equal to the number of information bits. In this case, since the number of information bits is 3, we need to write down the powers of x up to x^2. So we have:

x^0 = 1
x^1 = x
x^2

Step 3: Multiply each power of x by the code generator polynomial g(x), and write down the results in a matrix form. The coefficients of each term will be the columns of the generator matrix. In this case, we have:

For x^0:
1 * (1 + x^2) = 1 + x^2

For x^1:
x * (1 + x^2) = x + x^3

For x^2:
x^2 * (1 + x^2) = x^2 + x^4

Step 4: Since the degree of the generator polynomial g(x) is 2, we need to reduce the degrees of x in the matrix obtained in Step 3. We can use the property that x^3 = x and x^4 = x^2. So the reduced matrix is:

1 + x^2 -> 1 + x^2
x + x^3 -> x + x
x^2 + x^4 -> x^2 + x^2

Step 5: Finally, the matrix obtained in Step 4 is the generator matrix G for the (5,3) code with the given code generator polynomial g(x). So the generator matrix G is:

G = [1 + x^2, x + x, x^2 + x^2]

Therefore, the generator matrix G for the (5,3) code with the code generator polynomial g(x) = 1 + x^2 is:

G = [1 + x^2, x + x, x^2 + x^2]