The generator matrix

 1  0  0  0  1  1  1  0  X  1
 0  1  0  0  0  1  1  1  0  0
 0  0  1  0  1  1  0  1  0  0
 0  0  0  1  1  0  1  1  0  0
 0  0  0  0  X  0  0  0  X  0
 0  0  0  0  0  X  0  0  X  0
 0  0  0  0  0  0  X  0  X  0
 0  0  0  0  0  0  0  X  X  0

generates a code of length 10 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+84x^4+56x^5+232x^6+280x^7+462x^8+688x^9+504x^10+688x^11+420x^12+280x^13+280x^14+56x^15+57x^16+8x^18

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^2) for dimension 12.
This code was found by Heurico 1.16 in 0.0279 seconds.