An asymptotic property of quaternary additive codes

Let $n_k(s)$ be the maximal length $n$ such that a quaternary additive $[n,k,n-s{]}_4$-code exists. We solve a natural asymptotic problem by determining the lim sup ${\lambda }_k$ of $n_k(s)\mathrm{/}s,$ and the smallest value of $s$ such that $n_k(s)\mathrm{/}s={\lambda }_k\mathrm{.}$ Our new family of quaternary additive codes has parameters $[4^k-1,k,4^k-4^{k-1}{]}_4=[2^{2k}-1,k,3\cdot 2^{2k-2}{]}_4$ (where $k=l\mathrm{/}2$ and $l$ is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation meet the Griesmer bound with equality. The proof is in terms of multisets of lines in $PG(l-1,2)\mathrm{.}$