The generator matrix

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

generates a code of length 8 over F4[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+30x^16+60x^20+828x^24+84x^28+21x^32

The gray image is a linear code over GF(4) with n=32, k=5 and d=16.
This code was found by Heurico 1.16 in 0 seconds.