The generator matrix

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

generates a code of length 79 over F4[X]/(X^2) who´s minimum homogenous weight is 220.

Homogenous weight enumerator: w(x)=1x^0+375x^220+768x^221+324x^222+468x^223+1530x^224+2004x^225+768x^226+1092x^227+2313x^228+3444x^229+1212x^230+1320x^231+3012x^232+4788x^233+1764x^234+1608x^235+3357x^236+5340x^237+1920x^238+1740x^239+3177x^240+5160x^241+1512x^242+1572x^243+2961x^244+3564x^245+1260x^246+1056x^247+1860x^248+1908x^249+372x^250+336x^251+624x^252+612x^253+84x^254+24x^255+132x^256+60x^257+51x^260+33x^264+9x^268+12x^272+3x^276+3x^280+3x^284

The gray image is a linear code over GF(4) with n=316, k=8 and d=220.
This code was found by Heurico 1.16 in 28 seconds.