The generator matrix

 1  1  X  X  1
 0 X^3 X^3  0  0

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

Homogenous weight enumerator: w(x)=1x^0+1x^4+26x^5+2x^6+2x^7

The gray image is a linear code over GF(2) with n=40, k=5 and d=16.
This code was found by Heurico 1.16 in 6.87e-008 seconds.