On some covering designs

Let n ? k ? t be positive integers, and let Ω be a set of n elements. Let C(n, k, t) denote the number of k-tuples of Ω in a minimal system of k-tuples such that every t-tuple is contained in at least one k-tuple of the system. C(n, k, t) has been determined in all cases for which $\text{C(n, k, t) ?}\text{3(t + 1)}\text{2}$ W. H. Mills, Ars Combinatoria8 (1979), 199–315]. C(n, k, t) is determined in the case $\text{3(t + 1)}\text{2}\text{< C(n, k, t) ?}\text{3(t + 2)}\text{2}$.