The generator matrix

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

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+23x^2+44x^3+168x^4+552x^5+168x^6+44x^7+23x^8+1x^10

The gray image is a linear code over GF(2) with n=40, k=10 and d=8.
This code was found by Heurico 1.16 in 1.05e-007 seconds.