The generator matrix

 1  0  0  0  1
 0  1  0  0  1
 0  0  1  0  1
 0  0  0  1  1

generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+60x^2+240x^3+1030x^4+1440x^5+1020x^6+240x^7+65x^8

The gray image is a linear code over GF(2) with n=20, k=12 and d=4.
As d=4 is an upper bound for linear (20,12,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 12.
This code was found by Heurico 1.11 in -3.24e-008 seconds.