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.