New bounds for the language compression problem

The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD(A^⩽n) complexity of all strings x in some set A. The best known upper bound for this problem is 2log(||A^⩽n||)+O(log(n)), due to Buhrman and Fortnow. We show that the constant factor 2 in this bound is optimal. We also give new bounds for a certain kind of random sets R⊆{0, 1}ⁿ, for which we show an upper bound of log (||R^⩽n||)+O(log(n))