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 \(A, B\) 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 \(V\) be a vocabulary set (a set of "letters", in our previous case it was the set of natural numbers) , let \(n\) be a positive integer and \(W = V^n\) the set of n-length words (in our previous example \(n=6\) and \(W\) was the set of all possible 6-digits account numbers). Define for \(a,b \in W\) \(H(a,b)\) 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 \(a,b \in W\) , what is the chance that their distance is maxmised, i.e. that \(H(a,b)=n\) ? Generalising further, given \(d > 0\) what is the chance that \(k \geq 2\) words \(a_1,...,a_k \in W\) are such that \(H(a_i,a_j)\geq d \) for all \(1\leq a_i < a_j \leq k\)?
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!