the 3 isometry classes of irreducible [15,4,9]_3 codes are: code no 1: ================ 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0 2 2 2 2 1 1 1 1 0 0 0 0 2 0 0 2 2 1 0 2 2 1 0 1 1 0 0 0 2 0 2 1 2 1 2 0 0 1 2 1 0 0 0 0 2 the automorphism group has order 78 and is strongly generated by the following 4 elements: ( 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 2 2 2 2 2 2 2 2 2 2 2 , 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 , 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 2 2 2 2 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 2 0 1 1 2 0 2 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 , 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 2 0 1 1 2 0 2 2 0 2 2 2 2 1 1 1 1 0 0 0 1 2 1 2 1 0 0 2 1 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (11, 12), (2, 5, 3)(4, 6, 9)(7, 10, 8)(13, 14, 15), (1, 6, 10)(2, 5, 9)(3, 14, 7)(4, 8, 13), (1, 13, 4, 14, 3, 9, 6, 7, 10, 15, 5, 8, 2) orbits: { 1, 10, 2, 7, 6, 3, 9, 8, 14, 4, 5, 13, 15 }, { 11, 12 } code no 2: ================ 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0 2 2 2 2 1 1 1 1 0 0 0 0 2 0 0 2 2 1 0 2 2 1 0 1 1 0 0 0 2 0 2 0 2 1 2 1 2 0 1 0 1 0 0 0 2 the automorphism group has order 36 and is strongly generated by the following 4 elements: ( 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2 2 2 2 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 1 0 2 2 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 2 1 2 1 2 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 , 2 0 0 0 0 0 0 0 0 0 0 2 2 1 0 2 2 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 , 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 1 1 1 1 1 1 1 1 1 1 1 , 0 0 0 0 0 0 0 0 0 0 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 2 0 1 1 2 0 2 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (2, 8)(3, 13)(4, 12)(5, 14)(6, 11)(9, 15), (2, 5, 8, 14)(3, 15, 13, 9)(4, 6, 12, 11)(7, 10), (1, 5)(2, 6)(3, 7)(4, 8)(9, 10)(11, 12), (1, 6, 11)(2, 5, 12)(3, 7, 13)(4, 14, 8) orbits: { 1, 5, 11, 14, 2, 6, 12, 8, 4 }, { 3, 13, 9, 7, 15, 10 } code no 3: ================ 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0 2 2 2 2 1 1 1 1 0 0 0 0 2 0 0 2 2 1 0 2 2 1 0 1 1 0 0 0 2 0 1 0 2 2 1 0 2 2 1 0 1 0 0 0 2 the automorphism group has order 720 and is strongly generated by the following 4 elements: ( 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 2 0 1 1 2 0 2 2 0 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 , 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 2 1 0 2 2 1 0 1 1 0 0 0 2 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 2 2 2 2 1 1 1 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 , 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 1 1 2 0 1 1 2 0 2 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 , 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 1 1 1 1 1 1 1 1 1 1 1 ) acting on the columns of the generator matrix as follows (in order): (4, 14)(5, 12)(6, 11)(7, 13), (3, 4, 14)(5, 10, 12)(6, 9, 11)(7, 13, 8), (2, 5, 7, 10)(3, 14, 9, 6)(4, 11)(8, 15, 13, 12), (1, 6)(2, 5)(3, 7)(4, 8)(9, 10)(11, 12) orbits: { 1, 6, 11, 9, 4, 12, 14, 10, 3, 8, 5, 13, 7, 2, 15 }