The generator matrix

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

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+48x^70+432x^71+195x^72+192x^73+288x^74+1104x^75+174x^76+576x^77+768x^78+2736x^79+144x^80+1344x^81+1248x^82+3312x^83+147x^84+960x^85+720x^86+1632x^87+87x^88+108x^92+69x^96+27x^100

The gray image is a linear code over GF(4) with n=108, k=7 and d=68.
This code was found by Heurico 1.16 in 0.498 seconds.