Codes associated with the odd graphs

Linear codes arising from the row span over any prime field F"p of the incidence matrices of the odd graphs O"k for k>=2 are examined and all the main parameters obtained. A study of the hulls of these codes for p=2 yielded that for O"2 (the Petersen graph), the dual of the binary hull from an incidence matrix is the binary code from points and lines of the projective geometry PG"3(F"2), which leads to a correspondence between the edges and vertices of O"2 with the points and a collection of ten lines of PG"3(F"2), consistent with the codes. The study also gives the dimension, the minimum weight, and the nature of the minimum words, of the binary codes from adjacency matrices of the line graphs L(O"k).