the 8 isometry classes of irreducible [23,13,6]_2 codes are: code no 1: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 1 1 0 1 1 0 1 1 1 0 0 1 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 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 1 , 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 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 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 ) acting on the columns of the generator matrix as follows (in order): (1, 18)(4, 6)(5, 16)(7, 9)(8, 19)(10, 20)(11, 13)(12, 14)(15, 17)(22, 23), (1, 6, 18, 4)(5, 7, 16, 9)(8, 15, 19, 17)(10, 23, 20, 22)(11, 12, 13, 14) orbits: { 1, 18, 4, 6 }, { 2 }, { 3 }, { 5, 16, 9, 7 }, { 8, 19, 17, 15 }, { 10, 20, 22, 23 }, { 11, 13, 14, 12 }, { 21 } code no 2: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 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 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 } code no 3: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 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 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 } code no 4: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 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 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 } code no 5: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 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 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 } code no 6: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (1, 13)(2, 16)(3, 8)(4, 15)(6, 14)(7, 11)(9, 18)(12, 17)(20, 23)(21, 22) orbits: { 1, 13 }, { 2, 16 }, { 3, 8 }, { 4, 15 }, { 5 }, { 6, 14 }, { 7, 11 }, { 9, 18 }, { 10 }, { 12, 17 }, { 19 }, { 20, 23 }, { 21, 22 } code no 7: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 1 elements: ( 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 ) acting on the columns of the generator matrix as follows (in order): (1, 14, 9, 6)(3, 20)(4, 19, 12, 11)(5, 13, 22, 10)(7, 21, 18, 16)(8, 17, 23, 15) orbits: { 1, 6, 9, 14 }, { 2 }, { 3, 20 }, { 4, 11, 12, 19 }, { 5, 10, 22, 13 }, { 7, 16, 18, 21 }, { 8, 15, 23, 17 } code no 8: ================ 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 1 elements: ( 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 1 ) acting on the columns of the generator matrix as follows (in order): (1, 21, 6, 11)(2, 12, 8, 9)(3, 23, 10, 7)(4, 17, 13, 22)(5, 15, 16, 20) orbits: { 1, 11, 6, 21 }, { 2, 9, 8, 12 }, { 3, 7, 10, 23 }, { 4, 22, 13, 17 }, { 5, 20, 16, 15 }, { 14 }, { 18 }, { 19 }