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

generates a code of length 78 over F4[X]/(X^2) who´s minimum homogenous weight is 220.

Homogenous weight enumerator: w(x)=1x^0+156x^220+174x^224+48x^225+162x^228+432x^229+150x^232+1296x^233+105x^236+1296x^237+36x^240+54x^244+24x^248+51x^252+36x^256+36x^260+15x^264+6x^268+9x^272+3x^280+3x^284+3x^300

The gray image is a linear code over GF(4) with n=312, k=6 and d=220.
This code was found by Heurico 1.16 in 3.02 seconds.