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

generates a code of length 67 over F4[X]/(X^2) who´s minimum homogenous weight is 176.

Homogenous weight enumerator: w(x)=1x^0+54x^176+204x^180+411x^184+570x^188+1155x^192+2439x^196+4323x^200+4104x^204+1959x^208+330x^212+276x^216+192x^220+171x^224+111x^228+54x^232+18x^236+12x^240

The gray image is a linear code over GF(4) with n=268, k=7 and d=176.
This code was found by Heurico 1.16 in 3 seconds.