Binary codes and partial permutation decoding sets from the odd graphs

For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω{k}, the set of all k-subsets of Ω = {1, 2, …, 2k +1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k4 is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement $\overline {O(k)} $, is investigated.