# Expected value of minimum Hamming distance

#### billobillo

##### New member
The Hamming distance(HD) between two strings of equal length is the number of characters that differ between the two strings at the same position, for example the HD between "gold" and "wolf" is 2; the MHD is between N strings and is equal to the minimum of HD's among all possible combinations of size k.
My question is: If I randomly select k binary strings of length n out of N=2^n strings , what is the expected value of minimum Hamming distance(MHD) between the k selected strings?
I need to find a general formula that gives me the expected value of the MHD between k random strings of size n selected out of N where (N =2^n)

Below is a table that shows, for N=8 and all k's, the MHD plus its occurrences frequency:

 MHD OF k=2 k=3 k=4 k=5 k=6 k=7 k=8 1 12 48 68 56 28 8 1 2 12 8 2 3 4

Below another table for N=16

 MHD OF k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 ....... 1 32 352 1592 4240 7952 11424 12868 11440 2 48 208 228 128 56 16 2 3 32 4 8

There are more results if you need, I appreciate any help from you.
Regards.