B1=(({{1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0}, {1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1}, {0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1}, {1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0}, {1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1}, {1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1}, {0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0}, {0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0}, {1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1}}));

Transpose[B1]==B1

MatrixRank[B1]

B=(({{1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1}, {1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1}, {0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1}, {1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1}, {1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1}, {0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1}, {0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1}, {1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1}, {0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0}}));

MatrixRank[B]

Transpose[B]//MatrixForm
Transpose[B]==B

G=(({{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0}}));G1=(({{1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}}));

MatrixRank[G]

H=(({{1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1}, {1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1}, {0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1}, {1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1}, {1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1}, {0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1}, {0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1}, {1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1}, {0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}}));H1=(({{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1}, {1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1}, {0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1}, {1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1}, {1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1}, {0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1}, {0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1}, {1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1}, {0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0}}));

Mod[G.H,2]//MatrixForm

Mod[G1.H1,2]//MatrixForm

Mod[B.B,2]//MatrixForm