I have the following problem :
I generate n-bits long binaries (0|1), where every bit is 0 OR 1 with probability 50%. So, say I generate 'm' such binaries.
The question is what is the probability when generating the next one that it will be still ~50% away (by hamming distance) from all the generated vectors i.e. almost orthogonal.
said differently, how many binaries I can generate before I generate one that is closer than ~50% away.
Normally if 'n' is big it is almost guaranteed the binary vector is orthogonal to all previously generated. I want to find the threshold 'm' of number of vectors I can safely generate for defined 'n'.
I have hard time coming up with formulation on solution to the problem. So even if you don't have a solution, but just idea on how to mathematically formulate the problem, please comment out.
I was thinking along the lines of single bit logical operations (match i.e. XOR) extended somehow over multiple n-bits. Then probabilistically treating m-count of vectors in pairs... via at-least one match ..