Planes, Graphs and Codes#
Magma provides the following functions to create the graphs and codes naturally associated to a projective or affine plane.
- LineGraph(P): Plane -> Grph#
The line graph of the plane \(P\).
- IncidenceGraph(P): Plane -> Grph#
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 \(p \in V\) is adjacent to a vertex \(l \in L\) whenever \(p \in l\).
- LinearCode(P, K): Plane, FldFin -> Code#
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 \(K^{(v)}\).
- Example: codes (ex-43be98)#
> 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]