The generator matrix

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

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+30x^8+75x^12+165x^16+270x^20+3072x^21+300x^24+183x^28

The gray image is a linear code over GF(4) with n=28, k=6 and d=8.
This code was found by Heurico 1.16 in 0.00291 seconds.