The generator matrix

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

generates a code of length 4 over Z2[X]/(X^3) who´s minimum homogenous weight is 3.

Homogenous weight enumerator: w(x)=1x^0+20x^3+21x^4+20x^5+2x^6

The gray image is a linear code over GF(2) with n=16, k=6 and d=6.
As d=6 is an upper bound for linear (16,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 3.31e-005 seconds.