The Optimal Rate of MDS Variable Generation.

A collection of K random variables are called (K,n)-MDS if any n of the K variables are independent and determine all remaining variables. In the MDS variable generation problem, K users wish to generate variables that are (K,n)-MDS using a randomness variable owned by each user. We show that to generate 1 bit of (K,n)-MDS variables for each n ∈ {1,2, ⋯ ,K}, the minimum size of the randomness variable at each user is 1+1/2+ ⋯ +1/K bits.