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Proof

The Proof for $ q=2$ is simply that plugging $ q=2$ into the equation given in the definition of the Hamming code gives us that $ n=2^{r}-1$ which satisfies Theorem 1.4 which states that a perfect code has $ n$ of this form.

For $ q>2$ we need to generalize our previous GF(2) theorems a bit.

The metric still holds over GF($ q>2$). 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 $ a$ positions are identical. The result is the same.

The spheres S$ _{t}$(c) are mutually disjoint for GF($ q>2$). 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($ q>2$) 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 ($ 2t+1$ 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: $ \vert{\bf C}\vert(n*(q-1)+1)=q^{n}$.


next up previous
Next: Proof Up: Constructing G (and hence, Previous: Proof
Bill Cherowitzo 2001-12-11