The generator matrix

 1  0  0  1  1  1  1
 0  1  0  1 a^2*X a^2*X+1  0
 0  0  1 a^2  1  a  0
 0  0  0  X a^2*X a*X  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+60x^12+48x^13+336x^15+489x^16+144x^17+960x^18+2016x^19+1368x^20+3408x^21+2880x^22+3024x^23+1374x^24+240x^25+36x^28

The gray image is a linear code over GF(4) with n=28, k=7 and d=12.
This code was found by Heurico 1.16 in 0.031 seconds.