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.