The generator matrix

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

generates a code of length 3 over F4[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+9x^4+144x^6+27x^8+384x^9+432x^10+27x^12

The gray image is a linear code over GF(4) with n=12, k=5 and d=4.
This code was found by Heurico 1.16 in 3.24e-008 seconds.