The generator matrix

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

generates a code of length 5 over F4[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+15x^4+90x^8+270x^12+3072x^15+405x^16+243x^20

The gray image is a linear code over GF(4) with n=20, k=6 and d=4.
This code was found by Heurico 1.16 in 0.00164 seconds.