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Ball-avoiding randomness in an Hamming metric

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:

  • \(A: \) 21-33-45
  • \(B: \) 27-33-45

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!

Note by Ermes Trismegisto
4 months, 2 weeks ago

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Given any two words \(a,b \in W\) , what is the chance that their distance is maxmised, i.e. that \(H(a,b)=n\) ?

Isn't it \((V-1)^ n \) Lokesh Sharma · 4 months, 2 weeks ago

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@Lokesh Sharma Yeah, I agree with you that's pretty easy: you choose the letters the first word \(a\) and then when you choose the letters of \(b\) you can't use one character for each letter, so that one is farily easy. Ermes Trismegisto · 4 months, 2 weeks ago

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@Ermes Trismegisto

Generalising further, given \(d > 0\) what is the chance that \(k \geq 2\) words \(a_1,...,a_k \in W\) are such that \(H(a_i,a_j)\geq d \) ?

I assume both \(i\) and \(j\) can take any value from \(1\) to \(k\) except the case \( i = j \). Right? Lokesh Sharma · 4 months, 2 weeks ago

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@Lokesh Sharma Oh shit, good point ,yes! I'll edit it now, thanks! Ermes Trismegisto · 4 months, 2 weeks ago

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Hey there! Nice job on your first post! Welcome to the community.

I do not know a lot about Hamming Codes. Could you explain the question to me in some informal way? Agnishom Chattopadhyay · 4 months, 2 weeks ago

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@Agnishom Chattopadhyay Thank you very much! To be honest with you, I don't know much about them myself, just happened to know the name and learnt about metric space and it's a pretty popular example. I was a bit worried that I might have added some useless complications , so I'll try to rexpress the final question informally: basically, take \(k\) different \(n-\) letters words . (Here the words "letter" and "word" are to be intended in a more general mathematical fashion: for example £$$$%&??£ is a valid 9-letters word where each letter belong to the set of "keyboard keys". This set is called in general a vocabulary set and we can assume it to be finite: so your answer is also going to contain the number of element of this set (the cardinality of the set) as one of the parameters!).

So now you have \(k\) \(n-\)letter words.Given any two of these, you can count in how many characters or letters they differ: for example:

  • % % £ & & % $ %

  • % $ £ & & ? ? %

Those two words above are both 8 letters long and they differ in the 2nd,6th and 7th postion. We say that their Hamming distance is three, that is , they differ in three places, very easy!

Now you fix a positive number \(d\) and you ask what is the chance that taken any pair of the \(k\) words is such that their distance is at least \(d\), i.e. each pair ar at least \(d\) words "apart".

I hope this reformulation makes sense, let me know! Ermes Trismegisto · 4 months, 2 weeks ago

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@Ermes Trismegisto With you, so far.

So, we are trying to find the probability that a set of \(k\) \(n\) length words over some alphabet has the property that no two strings are closer than \(d\). Right? Agnishom Chattopadhyay · 4 months, 2 weeks ago

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@Agnishom Chattopadhyay Replace "closer" with "further apart" and that's precisely it. (\(H(a_i,a_j) \geq d\)) so we want the distance to be at least d. Ermes Trismegisto · 4 months, 2 weeks ago

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