This is an open ended question, I don't know the answer and would like your ideas on the topic.
Let me first give you a practical setting where this kind of problem arise: Banks need to issue account numbers for their customer. For example , let's say that are two customers of Tartaglia Bank and that they are issued the following 6-digits account numbers:
Now, it should be pretty evident that, by chance, the two account numbers happened to be pretty similar: they only differ in the second digit. This is not good! One digit is easily mistaken and A could possibly receive the money that someone inteded to send to B and viceversa.
It's clear that we would like any two random account numbers to be as different as possible, that is, in this case, to differ in all 6 digits.
This has now the following generalization, that is our actual question:
Let be a vocabulary set (a set of "letters", in our previous case it was the set of natural numbers) , let be a positive integer and the set of n-length words (in our previous example and was the set of all possible 6-digits account numbers). Define for to be the number of letters in which the two words differ (this is called an Hamming Distance and it can be shown to induce a Metric Space on W). Given any two words , what is the chance that their distance is maxmised, i.e. that ? Generalising further, given what is the chance that words are such that for all ?
I am fairly sure this problem or one of its variant should be fairly doable and I'll try to spend some time on it when I finish publishing these notes. It was just one of those real life inspired problems that I felt like to share on this platform as a way to test it (this is my first post here) and perhaps to interest some other people, so I am looking forward for your answers/ideas/generalisations/link to papers/critiques/suggestions.
Thanks for reading!