The generator matrix

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

generates a code of length 4 over F4[X]/(X^2) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+18x^8+192x^9+216x^12+576x^13+21x^16

The gray image is a linear code over GF(4) with n=16, k=5 and d=8.
This code was found by Heurico 1.16 in 3.24e-008 seconds.