The generator matrix

 1  1  X  1
 0  X  X  0
 0  0 X^2  0
 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+8x^2+32x^3+46x^4+32x^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 5.22e-005 seconds.