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 }