I hoped to present a projective geometry built upon GF(3) and a Hamming code embedded within as done for the Fano plane. I have been unable to reach this goal. But my research has shown the following results.
Construction techniques of Hamming codes in linear algebra allow corresponding geometric-space constructions.
Specifically, if we take each column of a parity check matrix H and let them correspond to a set S of points in a projective space PG(), then words of the dual code will correspond to complements of hyperplane sections in S.
I do not understand the above stated information.