The generator matrix

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

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+69x^4+272x^6+1082x^8+1248x^10+1082x^12+272x^14+69x^16+1x^20

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 0.0297 seconds.