Here's an example.

Let be a finite group of order generated by . Let denote the Cayley graph of and let be the incidence matrix of . Then the matrix is the parity check matrix of a LDPC code, provided is ``small'' with respect to . See [LR] and the references cited there for more details on such constructions.

Here's a small example using MAGMA. (Note that MAGMA does not have the ability to create a code from its parity check matrix, so we must treat the parity check matrix as the generator matrix of the dual code.)

> S3:=PermutationGroup< 3 | (1,2),(1,2,3)>; > Gamma:=CayleyGraph(S3); > Gamma; Digraph Vertex Neighbours 1 3 4 ; 2 3 4 ; 3 1 6 ; 4 2 5 ; 5 1 6 ; 6 2 5 ; > I:=IncidenceMatrix(Gamma); > I; [ 1 1 0 0 -1 0 0 0 -1 0 0 0] [ 0 0 1 1 0 0 -1 0 0 0 -1 0] [-1 0 -1 0 1 1 0 0 0 0 0 0] [ 0 -1 0 -1 0 0 1 1 0 0 0 0] [ 0 0 0 0 0 0 0 -1 1 1 0 -1] [ 0 0 0 0 0 -1 0 0 0 -1 1 1] > NumberOfColumns(I); 12 > NumberOfRows(I); 6 > M := KMatrixSpace(FiniteField(2), 6,12); > G:=M!I; > C1:=LinearCode(G); > C:=Dual(C1); > Dimension(C); 7 > MinimumDistance(C); 2 > C; [12, 7, 2] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 1 0 0 1 0 0 1] [0 1 0 0 0 0 0 1 1 0 0 0] [0 0 1 0 0 1 0 0 0 0 1 0] [0 0 0 1 0 0 0 1 0 0 1 1] [0 0 0 0 1 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 1 0 1]

David Joyner 2007-09-03