The generator matrix

 1  1  X  1  1
 0  X  X  0  0
 0  0 X^2  0 X^2
 0  0  0 X^3+X^2 X^2
 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+53x^2+48x^3+842x^4+2208x^5+842x^6+48x^7+53x^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 0 seconds.