The generator matrix

 1  1  1  1
 0  X  0  0
 0  0  X  0
 0  0  0  X

generates a code of length 4 over Z5[X]/(X^2) who´s minimum homogenous weight is 5.

Homogenous weight enumerator: w(x)=1x^0+16x^5+96x^10+256x^15+2500x^16+256x^20

The gray image is a linear code over GF(5) with n=20, k=5 and d=5.
This code was found by Heurico 1.16 in 0.000383 seconds.