1. Let's start by defining the minimum distance $d_{min}$ of a code. It is the smallest Hamming distance between any two distinct codewords in the code.\n\n2. The minimum distance determines the error detection and correction capabilities of the code. Specifically:\n- A code with minimum distance $d_{min}$ can detect up to $d_{min} - 1$ errors.\n- It can correct up to $t = \left\lfloor \frac{d_{min} - 1}{2} \right\rfloor$ errors, where $\lfloor x \rfloor$ denotes the floor function (greatest integer less than or equal to $x$).\n\n3. To understand why, consider that to detect $e$ errors, the received word must differ from any valid codeword by at least one position. Since the minimum distance is $d_{min}$, any error pattern affecting fewer than $d_{min}$ positions will not transform one codeword into another, so errors are detectable up to $d_{min} - 1$.\n\n4. For error correction, the code must be able to uniquely identify the original codeword despite errors. This requires that the spheres of radius $t$ around each codeword do not overlap. The largest $t$ satisfying this is $t = \left\lfloor \frac{d_{min} - 1}{2} \right\rfloor$.\n\n5. In summary, the key formulas are:\n$$\text{Error detection capability} = d_{min} - 1$$\n$$\text{Error correction capability} = \left\lfloor \frac{d_{min} - 1}{2} \right\rfloor$$\n\n6. These results are fundamental in coding theory and help design codes with desired error handling properties.