The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+12x^4+48x^6+54x^8+384x^9+288x^10+1644x^12+1152x^13+432x^14+81x^16

The gray image is a linear code over GF(4) with n=16, k=6 and d=4.
This code was found by Heurico 1.16 in 0.000864 seconds.