The generator matrix

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

generates a code of length 4 over Z2[X]/(X^3) who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+24x^1+220x^2+936x^3+1734x^4+936x^5+220x^6+24x^7+1x^8

The gray image is a linear code over GF(2) with n=16, k=12 and d=2.
As d=2 is an upper bound for linear (16,12,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 12.
This code was found by an older version of Heurico in 0 seconds.