The generator matrix

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

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+24x^12+165x^16+768x^18+396x^20+2304x^22+402x^24+36x^28

The gray image is a linear code over GF(4) with n=28, k=6 and d=12.
This code was found by Heurico 1.16 in 0.00238 seconds.