The generator matrix

 1  1  1  1  1  1  1
 0  X  0 X^2+X X^2 X^2+X X^2
 0  0 X^2 X^2 X^2  0  0

generates a code of length 7 over Z2[X]/(X^3) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+7x^6+48x^7+7x^8+1x^14

The gray image is a linear code over GF(2) with n=28, k=6 and d=12.
As d=12 is an upper bound for linear (28,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 0.000189 seconds.