The generator matrix

 1  1  1  1
 0  X  0 X^2
 0  0  X 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+8x^2+16x^3+78x^4+16x^5+8x^6+1x^8

The gray image is a linear code over GF(2) with n=16, k=7 and d=4.
This code was found by Heurico 1.16 in 4.22e-005 seconds.