Optimal Ternary Codes with Weight $w$ and Distance $2w-2$ in $\ell_1$-Metric

TL;DR

This paper determines the maximum size of certain ternary constant-weight codes in the -metric with specific parameters, advancing understanding in coding theory motivated by DNA data storage applications.

Contribution

It provides a complete characterization of the maximum size of ternary codes with weight w and distance 2w-2 for large n, using graph decomposition methods.

Findings

Maximum size of codes established for large n

Results extend previous sparse family cases

Applicable to DNA data storage error correction

Abstract

The study of constant-weight codes in $\ell_1$-metric was motivated by the duplication-correcting problem for data storage in live DNA. It is interesting to determine the maximum size of a code given the length $n$, weight $w$, minimum distance $d$ and the alphabet size $q$. In this paper, based on graph decompositions, we determine the maximum size of ternary codes with constant weight $w$ and distance $2w-2$ for all sufficiently large length $n$. Previously, this was known only for a very sparse family $n$ of density $4/w(w-1)$.