the 3 isometry classes of irreducible [8,3,4]_2 codes are: code no 1: ================ 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 0 1 0 0 0 1 the automorphism group has order 32 and is strongly generated by the following 4 elements: ( 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 , 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 , 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 , 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (5, 6), (3, 8)(4, 7)(5, 6), (3, 4)(7, 8), (1, 6, 2, 5)(3, 4) orbits: { 1, 5, 6, 2 }, { 3, 8, 4, 7 } code no 2: ================ 1 1 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 the automorphism group has order 384 and is strongly generated by the following 8 elements: ( 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 , 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 , 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 , 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 , 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 , 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 , 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 , 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 0 ) acting on the columns of the generator matrix as follows (in order): (5, 8), (4, 7), (4, 5, 7, 8), (3, 7, 6, 4), (3, 7, 8)(4, 5, 6), (1, 2), (1, 3, 2, 6), (1, 6, 5)(2, 3, 8)(4, 7) orbits: { 1, 2, 6, 5, 3, 8, 7, 4 } code no 3: ================ 1 1 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 the automorphism group has order 48 and is strongly generated by the following 4 elements: ( 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 , 1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 , 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 , 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (5, 8), (3, 6)(4, 7), (3, 4)(6, 7), (1, 4, 2, 7)(3, 6) orbits: { 1, 7, 4, 6, 2, 3 }, { 5, 8 }