Construction of m4-run linear graphs by finite geometries

Finite projective geometries are used to obtain some series of Taguchi's linear graphs involving m4 runs, where m is a prime or a prime power. The concept of maximal linear graph is introduced to reduce the number of nonisomorphic linear graphs. Some 81-run maximal linear graphs are given as examples. A table of 27 nonisomorphic 16-run maximal linear graphs, which is believed to be complete, is provided.