Is this coding exercise well thought?
I have to create a code C of 5 words with length , with the alphabet that corrects 1 mistake.
I am new to coding theory so I am having some troubles with this particular question...
If it corrects 1 mistakes, that means that (I believe this is a well-known result [I prove it at my courses]), where d is the minimal distance of the code C defined as , and that distance in the definition is the Hamming distance defined as , being and elements in (where is the alphabet of the code, and n denotes the length of the words).
So, in my particular case I want that the minimal distance of the code is greater than 2, i.e., I want that there are no two different words in the code that have Hamming distance equal or lesser than 2, i.e., no two words in the code that have more than 4 coordinates equal to each other (as If they have four equal to each other, there would be different to each other, so the Hamming distance would be 2, and if there are 5 equal to each other, the Hamming distance would be 1 which is not valid). So, I begin to construct the code, by taking the first word:
Then, I could take another one like:
which satisfies the previous conditions. So, taking this kind of words, and taking on account the conditions that must hold, I could keep going with this ''constructive algorithm'' until I have 5 words. For example:
And this would conclude the exercise (by taking all this 5 words listed above). I am not sure if this is a great argument and I would really appreciate if someone could give me some feedback on it...