The generator matrix

 1  1  X  X
 0 X^3 X^3  0

generates a code of length 4 over Z2[X]/(X^4) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+29x^4+2x^6

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^4) for dimension 5.
This code was found by Heurico 1.16 in 3.81e-009 seconds.