The generator matrix

 1  0  0  0  0  0  1  1  1  1
 0  1  0  0  0  0 X+1  0  X  0
 0  0  1  0  0  0  1  X  X  0
 0  0  0  1  0  0  1  X  0  0
 0  0  0  0  1  0  1  1 X+1  0
 0  0  0  0  0  1  1 X+1 X+1  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+65x^4+42x^5+242x^6+260x^7+469x^8+718x^9+496x^10+728x^11+463x^12+262x^13+250x^14+36x^15+58x^16+2x^17+4x^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 0.0292 seconds.