the 5 isometry classes of irreducible [16,13,4]_16 codes are:

code no       1:
================
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
3 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0
2 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0
8 4 1 0 0 0 1 0 0 0 0 0 0 0 0 0
9 5 1 0 0 0 0 1 0 0 0 0 0 0 0 0
13 6 1 0 0 0 0 0 1 0 0 0 0 0 0 0
12 7 1 0 0 0 0 0 0 1 0 0 0 0 0 0
14 8 1 0 0 0 0 0 0 0 1 0 0 0 0 0
15 9 1 0 0 0 0 0 0 0 0 1 0 0 0 0
7 10 1 0 0 0 0 0 0 0 0 0 1 0 0 0
6 11 1 0 0 0 0 0 0 0 0 0 0 1 0 0
11 12 1 0 0 0 0 0 0 0 0 0 0 0 1 0
10 13 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:
(
8 0 0 
0 8 0 
8 8 8 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(3, 4)(5, 6)(7, 8)(9, 10)(11, 12)(13, 14)(15, 16)
orbits: { 1 }, { 2 }, { 3, 4 }, { 5, 6 }, { 7, 8 }, { 9, 10 }, { 11, 12 }, { 13, 14 }, { 15, 16 }

code no       2:
================
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
3 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0
2 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0
8 4 1 0 0 0 1 0 0 0 0 0 0 0 0 0
9 5 1 0 0 0 0 1 0 0 0 0 0 0 0 0
13 6 1 0 0 0 0 0 1 0 0 0 0 0 0 0
12 7 1 0 0 0 0 0 0 1 0 0 0 0 0 0
14 8 1 0 0 0 0 0 0 0 1 0 0 0 0 0
7 10 1 0 0 0 0 0 0 0 0 1 0 0 0 0
6 11 1 0 0 0 0 0 0 0 0 0 1 0 0 0
11 12 1 0 0 0 0 0 0 0 0 0 0 1 0 0
10 13 1 0 0 0 0 0 0 0 0 0 0 0 1 0
4 14 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:
(
3 14 1 
7 13 3 
15 11 4 
, 2
)
acting on the columns of the generator matrix as follows (in order):
(1, 7)(2, 15)(3, 6)(4, 13)(5, 11)(8, 14)(9, 12)(10, 16)
orbits: { 1, 7 }, { 2, 15 }, { 3, 6 }, { 4, 13 }, { 5, 11 }, { 8, 14 }, { 9, 12 }, { 10, 16 }

code no       3:
================
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
3 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0
2 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0
8 4 1 0 0 0 1 0 0 0 0 0 0 0 0 0
9 5 1 0 0 0 0 1 0 0 0 0 0 0 0 0
13 6 1 0 0 0 0 0 1 0 0 0 0 0 0 0
12 7 1 0 0 0 0 0 0 1 0 0 0 0 0 0
14 8 1 0 0 0 0 0 0 0 1 0 0 0 0 0
7 10 1 0 0 0 0 0 0 0 0 1 0 0 0 0
11 12 1 0 0 0 0 0 0 0 0 0 1 0 0 0
10 13 1 0 0 0 0 0 0 0 0 0 0 1 0 0
4 14 1 0 0 0 0 0 0 0 0 0 0 0 1 0
5 15 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 2 elements:
(
0 8 0 
8 0 0 
13 11 8 
, 2
, 
13 8 4 
2 5 13 
12 13 9 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(1, 2)(3, 15)(4, 16)(5, 9)(6, 10)(7, 14)(8, 13)(11, 12), 
(1, 4, 5, 10, 12, 8, 15, 7)(2, 16, 9, 6, 11, 13, 3, 14)
orbits: { 1, 2, 7, 14, 15, 3, 8, 13, 12, 11, 10, 6, 5, 9, 4, 16 }

code no       4:
================
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
3 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0
5 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0
2 4 1 0 0 0 1 0 0 0 0 0 0 0 0 0
12 5 1 0 0 0 0 1 0 0 0 0 0 0 0 0
14 6 1 0 0 0 0 0 1 0 0 0 0 0 0 0
10 7 1 0 0 0 0 0 0 1 0 0 0 0 0 0
11 8 1 0 0 0 0 0 0 0 1 0 0 0 0 0
13 9 1 0 0 0 0 0 0 0 0 1 0 0 0 0
9 10 1 0 0 0 0 0 0 0 0 0 1 0 0 0
6 11 1 0 0 0 0 0 0 0 0 0 0 1 0 0
8 12 1 0 0 0 0 0 0 0 0 0 0 0 1 0
4 13 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 6 elements:
(
4 0 0 
0 14 0 
1 6 14 
, 2
, 
15 0 0 
2 7 6 
15 5 4 
, 3
, 
7 7 7 
3 7 2 
0 0 5 
, 2
, 
6 5 10 
5 2 13 
1 2 12 
, 0
, 
7 2 10 
3 0 0 
10 11 15 
, 1
, 
4 9 1 
3 1 14 
8 5 1 
, 3
)
acting on the columns of the generator matrix as follows (in order):
(3, 14)(4, 13)(7, 8)(9, 15)(10, 16), 
(2, 12, 11, 6)(3, 7, 9, 16)(8, 15, 10, 14), 
(1, 4)(2, 8)(6, 14)(10, 11)(12, 15), 
(1, 15)(2, 14)(3, 7)(4, 12)(6, 8)(9, 13)(10, 11), 
(1, 2, 12, 6)(3, 13, 7, 9)(4, 8, 15, 14), 
(1, 9, 11, 7)(2, 13, 12, 16)(3, 6)(4, 15, 10, 8)
orbits: { 1, 4, 15, 6, 7, 13, 12, 14, 8, 9, 11, 3, 2, 10, 16 }, { 5 }

code no       5:
================
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
3 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0
12 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0
8 5 1 0 0 0 1 0 0 0 0 0 0 0 0 0
15 6 1 0 0 0 0 1 0 0 0 0 0 0 0 0
9 7 1 0 0 0 0 0 1 0 0 0 0 0 0 0
6 8 1 0 0 0 0 0 0 1 0 0 0 0 0 0
4 9 1 0 0 0 0 0 0 0 1 0 0 0 0 0
13 10 1 0 0 0 0 0 0 0 0 1 0 0 0 0
14 11 1 0 0 0 0 0 0 0 0 0 1 0 0 0
11 12 1 0 0 0 0 0 0 0 0 0 0 1 0 0
2 13 1 0 0 0 0 0 0 0 0 0 0 0 1 0
7 14 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 960
and is strongly generated by the following 6 elements:
(
3 0 0 
0 5 0 
14 9 5 
, 2
, 
9 0 0 
0 8 0 
4 15 6 
, 3
, 
6 0 0 
11 14 13 
14 9 11 
, 1
, 
6 0 0 
12 15 2 
14 11 6 
, 1
, 
10 10 10 
6 5 12 
12 13 6 
, 2
, 
3 14 5 
6 10 8 
5 2 13 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(3, 12)(4, 13)(5, 6)(8, 15)(9, 14)(10, 11), 
(3, 10, 12, 11)(4, 5, 13, 6)(7, 16)(8, 9, 15, 14), 
(2, 7)(3, 15, 12, 8)(4, 11, 13, 10)(5, 14, 6, 9), 
(2, 6, 8, 12)(3, 14)(5, 9, 7, 15)(10, 13, 16, 11), 
(1, 5, 7, 12, 2, 4)(3, 10, 13, 15, 6, 9)(8, 11, 14), 
(1, 12, 3, 8)(2, 4, 6, 14)(5, 13, 11, 7)(9, 10)
orbits: { 1, 4, 8, 13, 6, 10, 2, 15, 14, 12, 3, 5, 11, 9, 7, 16 }