The generator matrix

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

generates a code of length 5 over Z2[X]/(X^4) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+41x^2+104x^3+726x^4+2352x^5+726x^6+104x^7+41x^8+1x^10

The gray image is a linear code over GF(2) with n=40, k=12 and d=8.
This code was found by Heurico 1.16 in -6.48e-008 seconds.