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

generates a code of length 44 over F4[X]/(X^2) who´s minimum homogenous weight is 120.

Homogenous weight enumerator: w(x)=1x^0+270x^120+192x^126+312x^128+1152x^130+1728x^134+210x^136+138x^144+66x^152+21x^160+6x^168

The gray image is a linear code over GF(4) with n=176, k=6 and d=120.
This code was found by Heurico 1.16 in 82.1 seconds.