Given a nonnegative sequence
of integers with Kraftsum at most 3/4, Ahlswede, Balkenhol and Khachatrian proposed the existence of a fix-free code with exactly
words for any length
. In this article complete thin fix-free codes are constructed and both so-called
-closed systems and multiplication are used to enlarge this class. In addition, a sufficient criterion is given in terms of elementary sequence-shifting preserving the fix-freedom of the associated code.