The generator matrix

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

generates a code of length 6 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+12x^4+16x^5+6x^6+16x^7+11x^8+2x^10

The gray image is a linear code over GF(2) with n=12, k=6 and d=4.
As d=4 is an upper bound for linear (12,6,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 6.
This code was found by Heurico 1.16 in 0.000129 seconds.