the 22 isometry classes of irreducible [15,6,7]_3 codes are: code no 1: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 0 2 1 0 1 0 1 0 0 0 0 2 0 2 0 1 2 1 2 2 0 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 2: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 0 2 1 0 1 0 1 0 0 0 0 2 0 2 0 1 2 1 2 2 1 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 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 0 1 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 1 0 1 2 0 2 0 2 1 1 2 2 2 2 0 0 0 2 0 1 2 1 2 2 1 1 0 0 1 0 0 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (2, 5)(3, 9)(6, 14)(7, 11)(8, 15)(12, 13) orbits: { 1 }, { 2, 5 }, { 3, 9 }, { 4 }, { 6, 14 }, { 7, 11 }, { 8, 15 }, { 10 }, { 12, 13 } code no 3: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 0 2 2 0 1 0 1 0 0 0 0 2 0 2 0 1 2 0 2 2 0 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 4: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 0 2 2 0 1 0 1 0 0 0 0 2 0 2 0 0 1 0 1 1 2 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 5: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 2 1 0 2 0 1 1 0 1 0 0 0 0 2 0 1 2 1 0 2 0 0 1 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 6: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 0 2 1 2 0 2 1 0 1 0 0 0 0 2 0 2 0 2 1 2 1 2 1 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 7: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 0 2 1 2 0 2 1 0 1 0 0 0 0 2 0 0 2 2 1 1 0 1 2 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 2 0 0 0 0 0 0 0 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 2 2 1 1 1 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1 2 2 0 2 1 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (1, 2)(4, 6)(5, 11)(7, 15)(10, 12)(13, 14) orbits: { 1, 2 }, { 3 }, { 4, 6 }, { 5, 11 }, { 7, 15 }, { 8 }, { 9 }, { 10, 12 }, { 13, 14 } code no 8: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 2 0 1 0 2 0 1 0 0 0 0 2 0 0 2 0 1 2 0 1 2 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 9: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 1 2 1 0 0 0 0 2 0 0 2 1 1 2 0 2 2 0 1 0 0 0 0 2 0 0 1 0 1 1 2 0 2 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 2 0 2 2 1 0 1 2 1 2 2 1 0 1 1 0 2 0 0 0 0 0 0 0 1 0 1 1 2 2 2 2 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 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (1, 15)(2, 14)(3, 8)(4, 11)(5, 9)(6, 7) orbits: { 1, 15 }, { 2, 14 }, { 3, 8 }, { 4, 11 }, { 5, 9 }, { 6, 7 }, { 10 }, { 12 }, { 13 } code no 10: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 0 2 2 1 1 0 1 0 1 0 0 0 0 2 0 2 0 1 0 2 1 1 2 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 11: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 2 1 0 2 1 0 1 0 1 0 0 0 0 2 0 0 0 2 1 0 1 2 1 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 12: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 0 2 1 2 1 0 1 0 1 0 0 0 0 2 0 2 1 0 2 2 0 1 2 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 13: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 0 2 2 1 0 1 1 0 1 0 0 0 0 2 0 0 0 1 0 1 2 2 1 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 14: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 2 0 2 1 1 0 1 0 0 0 0 2 0 0 1 1 2 0 2 2 1 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 2 1 2 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1 2 2 2 2 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 2 1 1 0 1 2 2 0 2 0 1 1 2 0 2 2 1 1 0 0 0 0 0 2 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (1, 12)(4, 11)(5, 10)(6, 9)(7, 14)(8, 15) orbits: { 1, 12 }, { 2 }, { 3 }, { 4, 11 }, { 5, 10 }, { 6, 9 }, { 7, 14 }, { 8, 15 }, { 13 } code no 15: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 2 0 2 1 1 0 1 0 0 0 0 2 0 2 2 1 0 2 0 0 2 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 16: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 1 2 0 2 2 1 1 0 1 0 0 0 0 2 0 0 2 0 1 2 0 2 1 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 17: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 0 2 1 2 1 0 0 0 0 2 0 0 2 1 0 2 1 0 2 0 1 0 0 0 0 2 0 2 2 1 0 1 2 2 1 1 0 0 0 0 0 2 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 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 } code no 18: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 0 2 0 1 2 1 2 1 0 0 0 0 2 0 0 0 2 2 0 2 1 1 0 1 0 0 0 0 2 0 1 0 0 1 2 0 1 2 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 1 0 0 0 0 0 0 0 0 0 2 0 1 2 1 2 1 0 2 2 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 2 1 0 2 1 2 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 2 2 0 2 0 0 0 0 0 1 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (2, 13)(3, 11)(5, 15)(6, 9)(7, 10)(8, 14) orbits: { 1 }, { 2, 13 }, { 3, 11 }, { 4 }, { 5, 15 }, { 6, 9 }, { 7, 10 }, { 8, 14 }, { 12 } code no 19: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 0 2 0 1 2 1 2 1 0 0 0 0 2 0 0 1 2 2 0 0 2 1 0 1 0 0 0 0 2 0 2 1 0 0 1 2 2 1 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 0 1 0 2 1 2 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 0 2 2 1 1 1 1 0 0 0 2 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 ) acting on the columns of the generator matrix as follows (in order): (1, 8)(3, 13)(4, 6)(7, 11)(9, 10)(14, 15) orbits: { 1, 8 }, { 2 }, { 3, 13 }, { 4, 6 }, { 5 }, { 7, 11 }, { 9, 10 }, { 12 }, { 14, 15 } code no 20: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 0 2 0 1 2 1 2 1 0 0 0 0 2 0 0 1 2 2 0 0 2 1 0 1 0 0 0 0 2 0 2 0 0 1 0 1 1 2 1 0 0 0 0 0 2 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 1 0 1 1 2 1 0 0 0 0 1 0 0 0 0 2 2 2 2 2 2 2 2 2 0 2 0 1 2 1 2 1 0 2 2 1 1 1 1 0 0 0 2 1 2 1 0 0 1 1 0 , 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 2 2 0 0 2 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 2 1 2 1 0 0 1 1 0 2 2 1 1 1 1 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (4, 15)(6, 10)(7, 13)(8, 11)(9, 12), (1, 3)(4, 7)(5, 14)(8, 12)(9, 11)(13, 15) orbits: { 1, 3 }, { 2 }, { 4, 15, 7, 13 }, { 5, 14 }, { 6, 10 }, { 8, 11, 12, 9 } code no 21: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 0 2 2 1 2 1 2 1 0 0 0 0 2 0 0 0 2 2 2 0 1 1 0 1 0 0 0 0 2 0 0 2 0 1 1 2 0 1 1 0 0 0 0 0 2 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 0 2 2 2 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 2 1 2 1 2 0 2 1 2 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (1, 14)(3, 13)(4, 12)(5, 9)(6, 7)(8, 10) orbits: { 1, 14 }, { 2 }, { 3, 13 }, { 4, 12 }, { 5, 9 }, { 6, 7 }, { 8, 10 }, { 11 }, { 15 } code no 22: ================ 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 2 2 1 1 1 1 0 0 0 0 2 0 0 0 0 2 1 2 1 0 0 1 1 0 0 0 2 0 0 0 2 1 0 2 1 0 1 0 1 0 0 0 2 0 0 2 0 1 0 2 0 2 1 1 0 0 0 0 2 0 0 1 1 1 0 2 2 2 1 0 0 0 0 0 2 the automorphism group has order 3 and is strongly generated by the following 1 elements: ( 0 1 1 1 0 2 2 2 1 1 1 1 1 1 1 1 1 1 1 0 2 0 1 0 1 2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 ) acting on the columns of the generator matrix as follows (in order): (1, 13, 15)(2, 11, 10)(3, 12, 14)(4, 6, 7)(5, 9, 8) orbits: { 1, 15, 13 }, { 2, 10, 11 }, { 3, 14, 12 }, { 4, 7, 6 }, { 5, 8, 9 }