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 = ∅. 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

PD-sets for the codes related to some classical varieties

AbstractWe generalize the notion of a PD-set of a code to that of a t-PD-set of an arbitrary permutation set. We find PD-sets for miquelian Benz planes of small order and for the ruled rational normal surface of order 3 in PG(4,3) and in PG(4,4). These results yield PD-sets for the related linear codes