The generator matrix

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

generates a code of length 7 over F4[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+45x^16+192x^18+108x^20+576x^22+90x^24+12x^28

The gray image is a linear code over GF(4) with n=28, k=5 and d=16.
This code was found by Heurico 1.16 in -1.01e-007 seconds.