The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+12x^10+2x^12+1x^16

The gray image is a linear code over GF(2) with n=20, k=4 and d=10.
As d=10 is an upper bound for linear (20,4,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.000224 seconds.