the 4 isometry classes of irreducible [8,4,3]_2 codes are: code no 1: ================ 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 the automorphism group has order 48 and is strongly generated by the following 4 elements: ( 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 , 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 , 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 , 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (4, 5), (2, 3)(4, 5)(6, 7), (2, 6)(3, 7), (1, 7, 8, 2)(3, 6) orbits: { 1, 2, 3, 6, 8, 7 }, { 4, 5 } code no 2: ================ 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 , 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 , 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (4, 5), (2, 3)(4, 5)(6, 7), (2, 6)(3, 7) orbits: { 1 }, { 2, 3, 6, 7 }, { 4, 5 }, { 8 } code no 3: ================ 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 the automorphism group has order 24 and is strongly generated by the following 5 elements: ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 , 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 , 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 , 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 , 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 ) acting on the columns of the generator matrix as follows (in order): (3, 4)(7, 8), (3, 7)(4, 8), (2, 3)(6, 7), (2, 6)(3, 7), (2, 7, 8)(3, 4, 6) orbits: { 1 }, { 2, 3, 6, 8, 4, 7 }, { 5 } code no 4: ================ 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 the automorphism group has order 8 and is strongly generated by the following 2 elements: ( 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 , 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ) acting on the columns of the generator matrix as follows (in order): (2, 7)(3, 6)(4, 5), (1, 7, 8, 2)(3, 5, 4, 6) orbits: { 1, 2, 7, 8 }, { 3, 6, 4, 5 }