Yes, it is always possible to achieve these exact bounds. This was done in a slightly different language in
Th. Skolem, "On Certain Distributions of Integers in Pairs with Given Differences", Mathematica Scandinavica 5 (1957), 57–68
For example, when $L=4k$ you can use the decomposition
- $(4k-2i,8k+i, 12k-i)$ for $0\le i\le 2k-1$
- $(4k-2i-1, 4k+i,8k-1-i)$ for $1\le i\le k-1$
- $(2k-2i-1, 5k+2+i,7k-1-i)$ for $0\le i\le k-3$
- $(1, 5k,5k+1)$ and $(4k-1, 6k+1, 10k)$