The generator matrix

 1  0  0  0  0  1  1  1  1  1
 0  1  0  0  0  0  0  1  X  0
 0  0  1  0  0  0  1  0  X  0
 0  0  0  1  0  1  1  1 X+1  0
 0  0  0  0  1  1  0  0  X  0
 0  0  0  0  0  X  0  X  X  0
 0  0  0  0  0  0  X  X  0  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+51x^4+90x^5+173x^6+300x^7+485x^8+654x^9+637x^10+584x^11+457x^12+342x^13+207x^14+76x^15+30x^16+2x^17+7x^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 0.0278 seconds.