The generator matrix

 1  0  0  0  1  1  1  1  1
 0  1  0  0  X X^2+1  1 X+1 X^2
 0  0  1  0 X+1  1  X X^3+1  0
 0  0  0  1  1  X X+1 X^3+1  X

generates a code of length 9 over Z2[X]/(X^4) who´s minimum homogenous weight is 5.

Homogenous weight enumerator: w(x)=1x^0+72x^5+636x^6+3864x^7+14454x^8+27464x^9+14496x^10+3848x^11+616x^12+80x^13+4x^14+1x^16

The gray image is a linear code over GF(2) with n=72, k=16 and d=20.
This code was found by Heurico 1.16 in 2.19 seconds.