The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 X 0 0 0 X X X 0 0 0 X 0 X X X
0 0 X 0 X X X 0 0 0 X X X X 0 0
0 0 0 X X 0 X X 0 X X 0 0 X X 0
generates a code of length 16 over Z2[X]/(X^2) who´s minimum homogenous weight is 16.
Homogenous weight enumerator: w(x)=1x^0+30x^16+1x^32
The gray image is a linear code over GF(2) with n=32, k=5 and d=16.
As d=16 is an upper bound for linear (32,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.00064 seconds.