the 8 isometry classes of irreducible [15,12,3]_4 codes are: code no 1: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 1 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 48 and is strongly generated by the following 4 elements: ( 1 0 0 0 1 0 0 1 1 , 1 , 1 0 0 0 1 0 1 1 1 , 1 , 3 0 0 1 2 2 2 2 0 , 0 , 3 0 0 3 2 2 1 2 0 , 0 ) acting on the columns of the generator matrix as follows (in order): (3, 11)(4, 8)(6, 7)(9, 13)(10, 12), (3, 4)(6, 7)(8, 11)(9, 12)(10, 13)(14, 15), (2, 10, 13)(3, 4, 5)(6, 9, 11)(7, 8, 12), (2, 9, 12)(3, 13, 7)(4, 6, 10)(5, 8, 11) orbits: { 1 }, { 2, 13, 12, 9, 10, 3, 8, 6, 11, 4, 5, 7 }, { 14, 15 } code no 2: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 36 and is strongly generated by the following 4 elements: ( 1 0 0 0 1 0 0 0 1 , 1 , 2 0 0 0 1 0 0 0 1 , 0 , 1 0 0 0 1 0 0 1 1 , 1 , 1 0 0 0 3 3 0 0 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (6, 7)(9, 10)(12, 13)(14, 15), (4, 13, 12)(5, 7, 6)(8, 10, 9), (3, 11)(4, 8)(6, 7)(9, 13)(10, 12), (2, 11)(4, 7)(5, 13)(6, 12)(8, 10) orbits: { 1 }, { 2, 11, 3 }, { 4, 12, 8, 7, 13, 10, 6, 9, 5 }, { 14, 15 } code no 3: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 1 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 1 0 2 2 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 12 and is strongly generated by the following 3 elements: ( 2 0 0 0 2 0 0 0 3 , 1 , 3 0 0 0 1 0 1 0 2 , 1 , 2 0 0 2 0 1 1 1 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 15)(6, 7)(8, 9)(11, 13)(12, 14), (3, 9)(4, 14)(5, 6)(11, 15)(12, 13), (2, 10)(3, 5)(6, 8)(7, 9)(11, 12)(13, 14) orbits: { 1 }, { 2, 10 }, { 3, 9, 5, 8, 7, 6 }, { 4, 15, 14, 11, 12, 13 } code no 4: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 1 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 3 0 0 3 0 2 2 2 0 , 1 , 0 1 3 0 1 0 3 3 0 , 1 , 0 1 0 0 1 3 3 3 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 10)(3, 5)(6, 8)(7, 9)(11, 12)(13, 14), (1, 15)(3, 5)(6, 13)(7, 11)(8, 14)(9, 12), (1, 10, 15, 2)(3, 5)(6, 9, 14, 11)(7, 8, 12, 13) orbits: { 1, 15, 2, 10 }, { 3, 5 }, { 4 }, { 6, 8, 13, 11, 14, 7, 12, 9 } code no 5: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 1 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 1 0 1 3 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 1 0 0 3 0 3 1 3 0 , 1 , 0 1 0 3 0 3 3 1 0 , 0 ) acting on the columns of the generator matrix as follows (in order): (2, 8)(3, 7)(5, 9)(6, 10)(11, 12), (1, 8, 2)(3, 4, 7)(5, 9, 14)(6, 10, 15)(11, 12, 13) orbits: { 1, 2, 8 }, { 3, 7, 4 }, { 5, 9, 14 }, { 6, 10, 15 }, { 11, 12, 13 } code no 6: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 0 3 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 1 0 2 3 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 720 and is strongly generated by the following 6 elements: ( 1 0 0 0 1 0 0 0 1 , 1 , 2 0 0 0 3 0 0 0 1 , 1 , 2 0 0 0 3 0 0 0 1 , 0 , 2 0 0 1 3 0 3 0 1 , 1 , 1 0 0 0 0 1 0 1 0 , 1 , 2 0 1 0 2 3 0 0 2 , 0 ) acting on the columns of the generator matrix as follows (in order): (6, 7)(9, 10)(12, 14)(13, 15), (4, 13)(5, 6)(8, 10)(11, 12), (4, 13, 15)(5, 6, 7)(8, 10, 9)(11, 12, 14), (2, 7)(3, 9)(4, 12)(11, 13), (2, 3)(5, 8)(6, 10)(7, 9), (1, 9)(2, 14)(4, 5)(6, 13)(8, 10)(11, 12) orbits: { 1, 9, 10, 3, 7, 8, 6, 2, 5, 13, 14, 4, 15, 11, 12 } code no 7: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 0 3 2 1 0 0 0 0 0 0 0 0 0 1 0 0 1 3 1 0 0 0 0 0 0 0 0 0 0 1 0 3 3 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 48 and is strongly generated by the following 4 elements: ( 2 0 0 0 3 0 0 0 1 , 1 , 2 0 0 0 3 0 1 0 1 , 0 , 3 0 0 3 0 3 3 3 0 , 1 , 2 0 0 3 3 0 1 0 1 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 13)(5, 6)(8, 10)(11, 12)(14, 15), (3, 10, 8)(4, 12, 15)(5, 6, 7)(11, 13, 14), (2, 8)(3, 5)(6, 9)(7, 10)(12, 13), (2, 6, 7, 5)(3, 10, 9, 8)(4, 13, 11, 12) orbits: { 1 }, { 2, 8, 5, 10, 9, 6, 7, 3 }, { 4, 13, 15, 12, 11, 14 } code no 8: ================ 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 1 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 1 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 1 0 0 0 3 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 1 0 2 3 1 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 360 and is strongly generated by the following 5 elements: ( 1 0 0 0 2 0 2 0 3 , 1 , 2 0 0 1 2 2 3 0 3 , 0 , 3 0 0 2 1 1 1 1 0 , 0 , 0 2 1 2 0 3 0 0 2 , 1 , 0 1 2 0 0 2 1 2 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (3, 8)(4, 12)(5, 6)(9, 11)(10, 13)(14, 15), (2, 13, 10)(3, 8, 7)(4, 6, 11)(5, 12, 9), (2, 8, 9)(3, 4, 5)(6, 7, 10)(11, 13, 12), (1, 14)(2, 8)(4, 5)(6, 13)(7, 11)(10, 12), (1, 8, 9, 14, 6, 11)(2, 5, 10, 4, 13, 3)(7, 15, 12) orbits: { 1, 14, 11, 15, 9, 6, 12, 7, 8, 5, 4, 10, 13, 3, 2 }