Determining properties of linear code from generator matrix
If the minimal distance is $d$, then


Up to $d-1$ errors can be detected. This means that if a sent codeword is modified by flipping at least one and up to $d-1$ bits, then the receiver can tell that some bits were flipped.


Indeed, suppose that $w$ is the codeword that was sent, and that $w'$ is obtained from $w$ by flipping at least one and up to $d-1$ bits. Since $1 \leq d_H(w,w') &lt; d$ (where $d_H$ is Hamming distance), we see that $w'$ is not a codeword (since the distance between any two different codewords is always at least $d$), and so we can detect that errors have occurred.


Up to $\lfloor \frac{d-1}{2} \rfloor$ can be corrected. This means that if a sent codeword is modified by flipping at most $\lfloor \frac{d-1}{2} \rfloor$ bits, then the original codeword can be recovered.


Indeed, suppose that $w$ is the codeword that was sent, and that $w'$ is obtained from $w$ by flipping up to $\lfloor \frac{d-1}{2} \rfloor$ bits. I claim that $w$ is the unique codeword at distance at most $\lfloor \frac{d-1}{2} \rfloor$ from $w'$. Indeed, if $z$ were another such codeword then
$$
d_H(w,z) \leq d_H(w,w') + d_H(w',z) \leq 2 \lfloor \tfrac{d-1}{2} \rfloor &lt; d,
$$
contrary to the definition of minimal distance.