I've been studying coding theory recently and most of the books I've looked at have had questions of the following type, but I haven't found any with a worked example:
Let C be a ternary code of block length 8 that can correct all single errors. Show that C cannot contain more than 25 code words.
I haven't been able to get my head around how to approach a question like this, other than to start with the fact that the minimum distance must be d>2. So there is a zero word, and the next word is of weight 3. But as for enumerating them, I feel a bit lost. I'm sure it comes down to basic combinatorics, but I just can't see it. Could anyone point me in the right direction, please?
You can easily construct more code words than that. For example consider the modulo $3$ sums
$$\sigma_0 = \sum s_j$$ $$\sigma_1 = \sum_{0,2,4,6}s_j$$ $$\sigma_2 = \sum_{0,1,4,5}s_j$$ $$\sigma_3 = \sum_{0,1,2,3}s_j$$
If you alter one ternit of $s_j$ you would alter $\sigma_0$ and also by the alteration of the other sums you could determine which ternit that was changed and the amount of change.
If you require the sums to all be zero you would get $81$ valid code words, and introducing an error on any of those you can by checking those sums identify which ternit that is wrong and correct it.