On the packing for triples

For positive integers $n\geq k\geq t$, a collection $\mathcal{B}$ of $k$-subsets of an $n$-set $X$ is called a $t$-packing if every $t$-subset of $X$ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and lower bounds for the maximum size of $3$-packings when $n$ is sufficiently larger than $k$. In one case, the upper and lower bounds are equal, in some cases, they differ by at most an additive constant depending only on $k$ and in one case they differ by a linear bound in $n$.