The generator matrix

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

generates a code of length 27 over F4[X]/(X^2) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+198x^64+351x^68+48x^69+441x^72+576x^73+558x^76+2592x^77+546x^80+5184x^81+696x^84+3888x^85+630x^88+426x^92+159x^96+81x^100+9x^104

The gray image is a linear code over GF(4) with n=108, k=7 and d=64.
This code was found by Heurico 1.16 in 0.99 seconds.