The generator matrix

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

generates a code of length 5 over F4[X]/(X^2) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+30x^12+192x^15+15x^16+18x^20

The gray image is a linear code over GF(4) with n=20, k=4 and d=12.
This code was found by Heurico 1.16 in -1.01e-007 seconds.