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 \(n_k(s)/s\) for s going to infinity, and the smallest value of s such that \(n_k(s)/s=\lambda _k.\) 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 \(k=l/2\) and l is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation with inner code \([3,2,2]_2\) meet the Griesmer bound with equality. The proof is in terms of multisets of lines in \(PG(l-1,2)\).