Optimal Ternary Constant-Weight Codes of Weight Four and Distance Six

Recently, Chee and Ling (“Constructions for $q$-ary constant-weight codes”, IEEE Trans. Inf. Theory, vol. 53, no. 1, 135–146, Jan. 2007 ) introduced a new combinatorial construction for $q$-ary constant-weight codes which reveals a close connection between $q$-ary constant-weight codes and sets of pairwise disjoint combinatorial designs. In this paper, we study the problem of constructing optimal ternary constant-weight codes with Hamming weight four and minimum distance six using this approach. The construction here exploits completely reducible super simple designs and group divisible codes. The problem is solved leaving only two cases undetermined. Previously, the sizes of constant-weight codes of weight four and distance six were known only for those of length no greater than 10.