The generator matrix

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

generates a code of length 7 over F4[X]/(X^2) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+15x^12+45x^16+90x^20+768x^21+66x^24+39x^28

The gray image is a linear code over GF(4) with n=28, k=5 and d=12.
This code was found by Heurico 1.16 in 1.01e-007 seconds.