The generator matrix

 1  1  1  1  1
 0 X^3  0  0  0
 0  0 X^3  0  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+5x^2+10x^4+224x^5+10x^6+5x^8+1x^10

The gray image is a linear code over GF(2) with n=40, k=8 and d=8.
This code was found by Heurico 1.16 in 0 seconds.