The generator matrix

 1  1  X  1  1
 0  X  X  0  0
 0  0 X^2  0 X^3
 0  0  0 X^3+X^2 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+25x^2+40x^3+358x^4+1200x^5+358x^6+40x^7+25x^8+1x^10

The gray image is a linear code over GF(2) with n=40, k=11 and d=8.
This code was found by Heurico 1.16 in 3.81e-009 seconds.