The generator matrix

 1  1  1  1
 0 X^2  0 X^2
 0  0 X^2 X^2

generates a code of length 4 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+30x^4+1x^8

The gray image is a linear code over GF(2) with n=16, k=5 and d=8.
As d=8 is an upper bound for linear (16,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 6.51e-005 seconds.