The generator matrix

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

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+16x^1+108x^2+432x^3+934x^4+432x^5+108x^6+16x^7+1x^8

The gray image is a linear code over GF(2) with n=16, k=11 and d=2.
This code was found by Heurico 1.16 in 0.000542 seconds.