Optimal Quaternary Constant-Weight Codes With Weight Four and Distance Five

Constant-weight codes play an important role in coding theory. The problem of determining the sizes for optimal quaternary constant-weight codes with length n, weight 4, and minimum Hamming distance 5 ( (n,5,4)₄ codes) has been investigated in several papers. Although some constructions and several infinite families for such codes with length n ≡ 0,1 mod 4 have been given, the problem is still far from complete. In this paper, we determine the size of an optimal (n,5,4)₄ code for each integer n ≥ 4 leaving 55 lengths unsolved. Especially, for length n ≡ 0,1 mod 4, the existence problem of the equivalent combinatorial object, namely the generalized Steiner system, is solved leaving only seven values undetermined.