the 16 isometry classes of irreducible [8,3,5]_5 codes are: code no 1: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 3 1 2 1 0 0 0 4 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 1 1 1 1 1 , 4 4 2 2 0 0 0 0 2 0 1 0 0 0 0 1 2 4 2 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (5, 6), (1, 3, 7)(2, 8, 4) orbits: { 1, 7, 3 }, { 2, 4, 8 }, { 5, 6 } code no 2: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 4 1 4 1 0 0 0 4 the automorphism group has order 12 and is strongly generated by the following 3 elements: ( 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 4 4 4 4 4 , 4 0 0 0 0 2 2 1 1 0 3 2 3 2 0 0 0 0 1 0 0 0 0 0 4 , 0 4 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 4 ) acting on the columns of the generator matrix as follows (in order): (5, 6), (2, 7)(3, 8), (1, 2)(3, 4) orbits: { 1, 2, 7 }, { 3, 8, 4 }, { 5, 6 } code no 3: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 3 2 1 0 1 0 0 4 the automorphism group has order 1 and is strongly generated by the following 0 elements: ( ) acting on the columns of the generator matrix as follows (in order): orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 } code no 4: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 4 2 1 0 1 0 0 4 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 4 4 4 4 4 ) acting on the columns of the generator matrix as follows (in order): (1, 2)(3, 4)(5, 6) orbits: { 1, 2 }, { 3, 4 }, { 5, 6 }, { 7 }, { 8 } code no 5: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 2 1 2 0 1 0 0 4 the automorphism group has order 4 and is strongly generated by the following 2 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 1 1 1 1 0 0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (2, 3)(4, 5)(7, 8), (1, 6)(2, 5)(3, 4) orbits: { 1, 6 }, { 2, 3, 5, 4 }, { 7, 8 } code no 6: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 3 1 2 0 1 0 0 4 the automorphism group has order 1 and is strongly generated by the following 0 elements: ( ) acting on the columns of the generator matrix as follows (in order): orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 } code no 7: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 3 2 2 0 1 0 0 4 the automorphism group has order 1 and is strongly generated by the following 0 elements: ( ) acting on the columns of the generator matrix as follows (in order): orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 } code no 8: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 4 3 2 0 1 0 0 4 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 4 3 2 0 1 0 1 0 0 0 0 0 0 0 3 0 0 0 4 0 0 0 2 0 0 , 0 0 1 0 0 0 2 0 0 0 1 2 3 0 4 1 1 1 1 1 3 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (1, 8)(3, 5), (1, 5, 8, 3)(4, 6) orbits: { 1, 8, 3, 5 }, { 2 }, { 4, 6 }, { 7 } code no 9: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 2 1 3 0 1 0 0 4 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 1 0 0 0 0 3 4 2 0 4 0 0 0 0 2 2 2 1 1 0 0 0 3 0 0 ) acting on the columns of the generator matrix as follows (in order): (2, 8)(3, 5)(4, 7) orbits: { 1 }, { 2, 8 }, { 3, 5 }, { 4, 7 }, { 6 } code no 10: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 3 1 3 0 1 0 0 4 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 0 0 2 0 3 3 3 3 3 0 0 2 0 0 2 0 0 0 0 0 0 0 0 2 ) acting on the columns of the generator matrix as follows (in order): (1, 4)(2, 6)(7, 8) orbits: { 1, 4 }, { 2, 6 }, { 3 }, { 5 }, { 7, 8 } code no 11: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 3 2 3 0 1 0 0 4 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 4 4 4 4 4 0 0 0 0 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): (1, 6)(2, 5)(3, 4) orbits: { 1, 6 }, { 2, 5 }, { 3, 4 }, { 7 }, { 8 } code no 12: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 4 2 3 0 1 0 0 4 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 3 3 3 3 3 4 4 2 2 0 0 0 3 0 0 1 3 2 0 4 0 0 0 0 3 ) acting on the columns of the generator matrix as follows (in order): (1, 6)(2, 7)(4, 8) orbits: { 1, 6 }, { 2, 7 }, { 3 }, { 4, 8 }, { 5 } code no 13: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 1 0 4 2 1 0 0 4 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 0 4 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 4 0 0 1 1 1 1 1 , 1 1 1 1 1 0 0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 4 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (1, 2)(3, 4)(5, 6), (1, 6)(2, 5)(3, 4) orbits: { 1, 2, 6, 5 }, { 3, 4 }, { 7 }, { 8 } code no 14: ================ 1 1 1 1 1 4 0 0 2 2 1 1 0 0 4 0 4 2 3 3 1 0 0 4 the automorphism group has order 8 and is strongly generated by the following 2 elements: ( 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 4 4 4 4 4 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (3, 4), (1, 6, 2, 5) orbits: { 1, 5, 2, 6 }, { 3, 4 }, { 7 }, { 8 } code no 15: ================ 1 1 1 1 1 4 0 0 3 2 1 1 0 0 4 0 1 2 3 0 1 0 0 4 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 ) acting on the columns of the generator matrix as follows (in order): (1, 3)(4, 5)(7, 8) orbits: { 1, 3 }, { 2 }, { 4, 5 }, { 6 }, { 7, 8 } code no 16: ================ 1 1 1 1 1 4 0 0 3 2 1 1 0 0 4 0 3 0 4 2 1 0 0 4 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 2 0 0 0 0 0 0 0 1 0 4 1 3 3 0 0 4 0 0 0 0 0 0 0 2 , 2 3 4 4 0 0 4 0 0 0 0 0 0 0 2 3 3 3 3 3 0 0 3 0 0 , 0 0 0 0 1 4 0 2 1 3 1 4 2 2 0 1 1 1 1 1 4 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (2, 4)(3, 7)(6, 8), (1, 7)(3, 5)(4, 6), (1, 5)(2, 8)(3, 7)(4, 6) orbits: { 1, 7, 5, 3 }, { 2, 4, 8, 6 }