The generator matrix
1 1 X 1
0 X X 0
0 0 X^2 0
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+6x^2+16x^3+18x^4+16x^5+6x^6+1x^8
The gray image is a linear code over GF(2) with n=16, k=6 and d=4.
This code was found by Heurico 1.16 in 4.72e-005 seconds.