The generator matrix

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

generates a code of length 27 over F4[X]/(X^2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+72x^68+195x^72+165x^76+768x^78+159x^80+2304x^82+123x^84+129x^88+99x^92+48x^96+21x^100+12x^104

The gray image is a linear code over GF(4) with n=108, k=6 and d=68.
This code was found by Heurico 1.16 in 0.0663 seconds.