The generator matrix

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

generates a code of length 8 over F4[X]/(X^2) who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+18x^20+222x^24+6x^28+9x^32

The gray image is a linear code over GF(4) with n=32, k=4 and d=20.
This code was found by Heurico 1.16 in -6.87e-008 seconds.