the 3 isometry classes of irreducible [8,3,5]_4 codes are: code no 1: ================ 1 1 1 1 1 1 0 0 2 2 1 1 0 0 1 0 3 2 1 0 1 0 0 1 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 2 0 0 0 0 0 2 0 0 0 2 2 2 2 2 0 0 0 0 2 0 0 0 2 0 , 1 , 0 2 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 2 0 0 2 2 2 2 2 , 0 ) acting on the columns of the generator matrix as follows (in order): (3, 6)(4, 5), (1, 2)(3, 4)(5, 6) orbits: { 1, 2 }, { 3, 6, 4, 5 }, { 7 }, { 8 } code no 2: ================ 1 1 1 1 1 1 0 0 2 2 1 1 0 0 1 0 2 1 2 0 1 0 0 1 the automorphism group has order 12 and is strongly generated by the following 3 elements: ( 2 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 2 0 , 0 , 0 0 0 2 0 0 0 2 0 0 2 2 2 2 2 0 0 0 0 2 2 0 0 0 0 , 0 , 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 3)(4, 5)(7, 8), (1, 5, 4)(2, 6, 3), (1, 6)(2, 5)(3, 4) orbits: { 1, 4, 6, 5, 3, 2 }, { 7, 8 } code no 3: ================ 1 1 1 1 1 1 0 0 2 2 1 1 0 0 1 0 3 1 3 0 1 0 0 1 the automorphism group has order 48 and is strongly generated by the following 5 elements: ( 2 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 2 0 , 1 , 3 0 0 0 0 0 0 0 0 1 1 2 1 0 2 0 0 1 0 0 1 1 3 3 0 , 0 , 0 2 0 0 0 2 0 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 2 0 0 , 1 , 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 , 1 , 2 2 3 3 0 0 0 0 3 0 0 0 1 0 0 0 3 0 0 0 0 0 0 0 1 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 3)(4, 5)(7, 8), (2, 7, 5)(3, 4, 8), (1, 2)(3, 5)(4, 6), (1, 6)(2, 5)(3, 4), (1, 7)(2, 4)(6, 8) orbits: { 1, 2, 6, 7, 3, 5, 4, 8 }