Optimal Codes With Small Constant Weight in ℓ₁-Metric

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in l ₁ -metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes with l ₁ -weight w ≤ 4 for all possible distances. In general, we derive the size of the largest ternary code with constant weight w and distance 2w-2 for sufficiently large length n satisfying n ≡ 1, w,- w+2,-2w+3 mod w(w-1).