The generator matrix

 1  0  1  1  1
 0  1  1  a  0
 0  0 a^2*X  0  X
 0  0  0  X a*X

generates a code of length 5 over F4[X]/(X^2) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+30x^8+48x^9+636x^12+288x^13+768x^15+1833x^16+432x^17+60x^20

The gray image is a linear code over GF(4) with n=20, k=6 and d=8.
This code was found by Heurico 1.16 in 0.00145 seconds.