[Next] [Prev] [_____] [Left] [Up] [Index] [Root]
Planes, Graphs and Codes

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 Plane_codes (H57E18)

> P, V, L := AffinePlane(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]

[Next] [Prev] [_____] [Left] [Up] [Index] [Root]