The generator matrix

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

generates a code of length 4 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+28x^2+198x^4+28x^6+1x^8

The gray image is a linear code over GF(2) with n=16, k=8 and d=4.
As d=5 is an upper bound for linear (16,8,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 8.
This code was found by Heurico 1.16 in 7.2e-005 seconds.