Ternary codes from graphs on triples

For a set Ω of size n ≥ 7 and Ω { 3 } the set of subsets of Ω of size 3 , we examine the ternary codes obtained from the adjacency matrix of each of the three graphs with vertex set Ω { 3 } , with adjacency defined by two vertices as 3 -sets being adjacent if they have zero, one or two elements in common, respectively.