What is the formula for determining how many errors a generator matrix can correct?

Question

I am wondering whether or not there is a generic formula for determining how many errors a generator matrix is able to correct if also provided the field the code is in. For example, given the following generator matrix applied to a code in F4:

\begin{bmatrix}1 & 0 & 0 & 2 & 1 & 0\\0 & 1 & 0 & 3 & 3 & 2\\0 & 0 & 1 & 0 & 0 & 3\end{bmatrix}

Is there a way to figure out how many errors it can correct?

Answer 1

TL; DNR version: It is a nontrivial task to determine how many errors can be corrected.

Generator matrices do not correct any errors; codes with appropriate decoding algorithms do, and not all decoding algorithms can correct all the errors that a code is capable of correcting. Be that as it may, determining the minimum distance of a code (which tells you how many errors the code is capable of correcting -- whether or not there is a decoding algorithm capable of correcting that many errors is a separate issue) is nontrivial. See the paper by Alexander Vardy "The intractability of computing the minimum distance of a code," IEEE Transactions on Information Theory, pp. 1757-1766, November 1997 which shows that the problem is NP-hard even for binary linear codes.

Answer 2

Since your generator matrix is in standard form, you can find the parity-check matrix easily. Then, by observing the columns, you can detect the minimum distance of this code. Suppose it is d. Then you can correct up to d/2 errors.