The generator matrix

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

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+41x^2+120x^3+406x^4+912x^5+406x^6+120x^7+41x^8+1x^10

The gray image is a linear code over GF(2) with n=20, k=11 and d=4.
As d=5 is an upper bound for linear (20,11,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 11.
This code was found by Heurico 1.11 in 0 seconds.