Short note: Error detection and correction capability

The error detection and correction capability of a block code depends on its minimum Hamming distance $d_{min}$, the smallest number of bit positions in which any two codewords differ. For a linear code, $d_{min}$ equals the smallest weight of a non-zero codeword.

Detection

If fewer than $d_{min}$ bits are changed, a codeword cannot turn into another valid codeword, so the error is noticed:

$$s\le d_{min}-1$$

Correction

The decoder chooses the nearest codeword. This is always right if the received word is closer to the sent codeword than to any other, which needs

$$t\le\left\lfloor\frac{d_{min}-1}{2}\right\rfloor\quad(d_{min}\ge 2t+1)$$

To correct $t$ and also detect $s>t$ errors: $d_{min}\ge t+s+1$.

  c1 o----x----x----o c2     d_min = 3
       radius t = 1 spheres do not overlap

Larger $d_{min}$ needs more redundancy (lower code rate $k/n$).