The generator matrix

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

generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+66x^4+56x^6+5x^8

The gray image is a linear code over GF(2) with n=20, k=7 and d=8.
As d=8 is an upper bound for linear (20,7,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 7.
This code was found by Heurico 1.16 in 0.000253 seconds.