An asymptotic property of quaternary additive codes

Let \(n_k(s)\) $n_k(s)$ be the maximal length n such that a quaternary additive \([n,k,n-s]_4\) ${[n,k,n-s]}_4$ -code exists. We solve a natural asymptotic problem by determining the lim sup \(\lambda _k\) ${\lambda }_k$ of \(n_k(s)/s\) $n_k(s)/s$ for s going to infinity, and the smallest value of s such that \(n_k(s)/s=\lambda _k.\) $n_k(s)/s={\lambda }_k.$ Our new family of quaternary additive codes has parameters \([4^k-1,k,4^k-4^{k-1}]_4=[2^{2k}-1,k,3\cdot 2^{2k-2}]_4\) ${[4^k-1,k,4^k-4^{k-1}]}_4={[2^{2k}-1,k,3·2^{2k-2}]}_4$ (where \(k=l/2\) $k=l/2$ and l is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation with inner code \([3,2,2]_2\) ${[3,2,2]}_2$ meet the Griesmer bound with equality. The proof is in terms of multisets of lines in \(PG(l-1,2)\) $PG(l-1,2)$ .