Magma provides the following functions to create the graphs and codes naturally associated to a projective or affine plane.
The line graph of the plane P.
The incidence graph of the plane P. This bipartite graph has as vertex set the union of the point set V and line set L of P. A vertex p ∈V is adjacent to a vertex l ∈L whenever p ∈l.
Given a plane P with v points and a finite field K, this function returns the linear code C of length v generated by the characteristic functions of the lines of P considered as vectors of the K-space K(v).
> P, V, L := FiniteAffinePlane(3); > #V, #L; 9 12 > IncidenceGraph(P); Graph Vertex Neighbours 1 10 11 12 19 ; 2 16 17 18 19 ; 3 13 14 15 19 ; 4 10 15 17 21 ; 5 12 14 16 21 ; 6 11 13 18 21 ; 7 10 14 18 20 ; 8 11 15 16 20 ; 9 12 13 17 20 ; 10 1 4 7 ; 11 1 6 8 ; 12 1 5 9 ; 13 3 6 9 ; 14 3 5 7 ; 15 3 4 8 ; 16 2 5 8 ; 17 2 4 9 ; 18 2 6 7 ; 19 1 2 3 ; 20 7 8 9 ; 21 4 5 6 ; > LinearCode(P, Field(P)); [9, 6, 3] Linear Code over GF(3) Generator matrix: [1 0 0 0 0 1 0 1 0] [0 1 0 0 0 1 0 2 2] [0 0 1 0 0 1 0 0 1] [0 0 0 1 0 2 0 1 2] [0 0 0 0 1 2 0 2 1] [0 0 0 0 0 0 1 1 1]