Find the linear block code generator matrix G, if the code generator polynomial is g(x) = 1 + x^3 for a (7,4) code.
To find the generator matrix, we need to determine the code's parity check matrix and then use its complement to construct the generator matrix.
Step 1: Determine the Parity Check Matrix H
The parity check matrix H for an (n, k) linear block code can be obtained by considering the coefficients of the code generator polynomial g(x) and forming a matrix from them.
Since the code generator polynomial is g(x) = 1 + x^3, we have the following coefficients: [1, 0, 0, 1].
To form the parity check matrix H, we need to write down the coefficients as rows of the matrix and then perform some row operations to obtain a suitable form. Let's do that:
[1, 0, 0, 1]
Now, we need to find a suitable form for this matrix. One common technique is to transform the matrix to its systematic form, which places the identity matrix at the rightmost end. However, since this is a (7, 4) code, and 4 does not evenly divide 7, we cannot form a systematic parity check matrix.
Instead, we can rearrange the rows to obtain the following form:
[1, 0, 0, 1]
Step 2: Find the Complement of H to Obtain G
To find the generator matrix G, we need to take the complement of the parity check matrix H by swapping 1s and 0s in each row. The complement of the matrix is denoted as H'.
After swapping 1s and 0s in each row of H, we get:
[0, 1, 1, 0]
This complement of H will be the generator matrix G for the given code.
So, the generator matrix G for the (7, 4) code, with the code generator polynomial g(x) = 1 + x^3, is:
G = [0, 1, 1, 0]