Binary codes and partial permutation decoding sets from the odd graphs
Abstract
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 = ∅. 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 k
4
is determined. Lastly,
the relationship between the dual code from O(k) and the code from its graph-theoretical complement O(k), is
investigated.