New results on ternary constant weight codes under ${\boldsymbol\ell}_{\boldsymbol1}$-metric

Abstract

Constant weight codes under $\ell_1$-metric are important in the technology of storage in live DNA. In this paper, we study the size of optimal ternary constant weight codes with given length $n$, $\ell_1$-weight $w$ and minimum $\ell_1$-distance $2w-4$. For general $n$ and $w$, we give an upper bound on the number of codewords in such a code. When $w=6$, we further improve the upper bound by computing the cost, and give lower bounds by constructions. In particular for all odd $n\not\equiv~9,~13,~17~({\rm~mod~}20)$, we determine the maximum number of codewords, or determine the first- or the second-order coefficients of $n$ asymptotically in the maximum number formula.