Suppose that I was asked to find the generator matrix for a $\left[4,2,2\right]$ code.

My first guess would be: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}$$

Since I know that the generator matrix is of the form: $$G = \left[ I_k | P \right]$$

And since the rows are linearly independent and a distance of $2$ apart, I would assume that this is a generator matrix.

The solution that I am given is that the $\textbf{standard}$ form of the generator matrix for $\left[4,2,2\right]$ is:

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

Which confuses me because it would appear that even though it forms a basis, the code words are a distance 4 apart.

I guess that I'm confused as to how to generate a generator matrix, some help would be greatly appreciated.


As far as the distance in the solution gets:

There is a codeword $C(11)=(1,0,1,0)+(0,1,0,1)=(1,1,1,1)$ which is 2 away from $(1,0,1,0)=C(01)$. The distance is the minimum over pairs in the subspace not all pairs of basis elements.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.