the 5 isometry classes of irreducible [16,13,3]_4 codes are: code no 1: ================ 1 1 1 1 0 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 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 1 0 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 0 1 0 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 72 and is strongly generated by the following 4 elements: ( 1 0 0 0 1 0 1 1 1 , 1 , 3 0 0 0 2 0 1 2 2 , 1 , 2 0 0 2 0 3 3 3 3 , 0 , 3 0 0 3 2 2 1 2 0 , 0 ) acting on the columns of the generator matrix as follows (in order): (3, 4)(6, 7)(8, 11)(9, 12)(10, 13)(14, 15), (3, 12)(4, 8)(5, 7)(9, 11)(10, 13)(14, 16), (2, 13, 10)(3, 5, 4)(6, 11, 9)(7, 12, 8), (2, 9, 12)(3, 13, 7)(4, 6, 10)(5, 8, 11) orbits: { 1 }, { 2, 10, 12, 13, 6, 9, 3, 7, 4, 11, 5, 8 }, { 14, 15, 16 } code no 2: ================ 1 1 1 1 0 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 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 1 0 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 0 1 0 0 3 1 0 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: ( 1 0 0 0 1 0 0 1 1 , 1 , 3 0 0 3 0 2 2 2 0 , 1 , 2 0 0 2 3 3 1 0 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (3, 11)(4, 8)(6, 7)(9, 13)(10, 12), (2, 10)(3, 5)(6, 8)(7, 9)(11, 12)(14, 15), (2, 12)(3, 10)(4, 7)(5, 11)(6, 13)(14, 15) orbits: { 1 }, { 2, 10, 12, 3, 11, 5 }, { 4, 8, 7, 6, 9, 13 }, { 14, 15 }, { 16 } code no 3: ================ 1 1 1 1 0 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 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 2 2 1 0 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 0 1 the automorphism group has order 16 and is strongly generated by the following 4 elements: ( 2 0 0 0 2 0 0 0 3 , 1 , 2 0 0 2 0 1 1 1 0 , 1 , 0 3 1 0 3 0 2 2 0 , 0 , 2 0 1 0 2 1 0 0 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 15)(6, 7)(8, 9)(11, 13)(12, 14), (2, 10)(3, 5)(6, 8)(7, 9)(11, 12)(13, 14), (1, 16)(3, 5)(4, 15)(6, 11)(7, 13)(8, 12)(9, 14), (1, 10)(2, 16)(4, 15)(6, 14)(7, 12) orbits: { 1, 16, 10, 2 }, { 3, 5 }, { 4, 15 }, { 6, 7, 8, 11, 14, 9, 13, 12 } code no 4: ================ 1 1 1 1 0 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 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 1 0 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 0 1 0 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 120 and is strongly generated by the following 4 elements: ( 1 0 0 1 0 3 3 3 0 , 1 , 1 0 0 1 3 0 3 0 3 , 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), (2, 5, 6, 7)(3, 10, 9, 8)(11, 12)(13, 15, 14, 16), (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, 13, 10, 7, 14, 16, 6, 12, 3, 9, 11, 8, 5 }, { 4 } code no 5: ================ 1 1 1 1 0 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 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 1 0 0 0 0 3 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 1 0 0 2 3 1 0 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 0 1 the automorphism group has order 5760 and is strongly generated by the following 9 elements: ( 2 0 0 0 3 0 0 0 3 , 1 , 2 0 0 0 3 0 3 0 1 , 1 , 1 0 0 0 1 0 2 0 2 , 0 , 3 0 0 0 3 0 3 3 1 , 0 , 3 0 0 2 1 1 1 1 0 , 0 , 2 0 0 1 2 2 3 0 3 , 0 , 0 2 1 2 0 3 0 0 2 , 1 , 0 1 2 0 0 2 1 2 3 , 1 , 1 1 2 1 3 2 3 1 3 , 0 ) acting on the columns of the generator matrix as follows (in order): (4, 9)(5, 6)(7, 8)(11, 14)(12, 15)(13, 16), (3, 8)(4, 12)(5, 6)(9, 11)(10, 13)(14, 15), (3, 8, 7)(4, 11, 15)(9, 12, 14)(10, 13, 16), (3, 4, 16)(7, 10, 14)(8, 9, 15)(11, 12, 13), (2, 8, 9)(3, 4, 5)(6, 7, 10)(11, 13, 12), (2, 13, 10)(3, 8, 7)(4, 6, 11)(5, 12, 9), (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), (1, 11, 14, 12, 3, 9, 15, 13, 2, 5, 10, 6, 8, 7, 16) orbits: { 1, 14, 11, 16, 15, 12, 10, 9, 4, 13, 6, 7, 5, 8, 3, 2 }