The generator matrix

 1  1  1  1  1
 0  X  0  0  X
 0  0  X  0 a^2*X
 0  0  0  X  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+60x^12+768x^15+105x^16+60x^20

The gray image is a linear code over GF(4) with n=20, k=5 and d=8.
This code was found by Heurico 1.16 in 6.87e-008 seconds.