For we need to generalize our previous GF(2) theorems a bit.
The metric still holds over GF(). The proof is the same as the one earlier, except when we prove the triangle identity. We must throw in one extra count that corresponds to when the positions of v do not match either u nor w when u's and w's positions are identical. The result is the same.
The spheres S(c) are mutually disjoint for GF(). The proof is identical to that of Theorem 1.3.
The minimum distance of C equals the minimum weight (defined the same way for GF() of C). The proof is identical to the earlier one.
For example, the Hamming code is a 1-error code. Since it's linear, we must have the zero vector present in the code. And since the least allowable distance between codewords is 3 ( from definition of code), we have its minimum weight is 3.
A linear 1-correcting code is perfect if and only if the following equation holds: .