The generator matrix

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

generates a code of length 3 over F4[X]/(X^2) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+36x^6+9x^8+96x^9+108x^10+6x^12

The gray image is a linear code over GF(4) with n=12, k=4 and d=6.
This code was found by Heurico 1.16 in 6.87e-008 seconds.