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

generates a code of length 68 over F4[X]/(X^2) who´s minimum homogenous weight is 180.

Homogenous weight enumerator: w(x)=1x^0+93x^180+279x^184+480x^188+663x^192+1473x^196+2934x^200+4173x^204+3720x^208+1518x^212+300x^216+213x^220+204x^224+174x^228+96x^232+30x^236+24x^240+6x^244+3x^248

The gray image is a linear code over GF(4) with n=272, k=7 and d=180.
This code was found by Heurico 1.16 in 3.08 seconds.