The generator matrix

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

generates a code of length 7 over F4[X]/(X^2) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+63x^8+210x^12+735x^16+1260x^20+12288x^21+1281x^24+546x^28

The gray image is a linear code over GF(4) with n=28, k=7 and d=8.
This code was found by Heurico 1.16 in 0.0434 seconds.