The generator matrix

 1  0  0  1  1  1  1  1  1  1  1  1 (a+1)X  1  1  1  1  0 (a+1)X  1  1  1  1  1  1  0  1  X  1  0  1  1  1  1  1  1  1  1  0 (a+1)X  X  1  X  1  1 aX  1  1 (a+1)X  1  1  X  1  1  1  1 aX (a+1)X  1  1  X  1  1  1  1  0  1  1 aX  1  1  1 (a+1)X  1  1  1  1  1  1
 0  1  0  0  0 (a+1)X  1 (a+1)X+a a+1 (a+1)X+1  a  a  1 (a+1)X+a (a+1)X+1 a+1 (a+1)X+a+1  1  1 (a+1)X+a+1 X+1 X+1 (a+1)X+1 X+a a+1  1 (a+1)X+a  0 (a+1)X+a+1  1 (a+1)X (a+1)X+a+1 (a+1)X  a  1 aX+a  a aX+a+1  1  1  1 aX  1 aX+1  X  1  1 aX  1 aX+1 a+1  1 a+1 aX+1 aX (a+1)X (a+1)X  1 aX+a+1  1  1 (a+1)X+a+1 (a+1)X+a  a X+a  1 a+1  X (a+1)X X+a  0  1 (a+1)X X+1 (a+1)X+a+1 aX aX+a aX+1 aX+a
 0  0  1  1  a a+1 X+a+1 a+1 X+a+1  a X+a  0 a+1 X+1  0 X+1 aX  a  1  a (a+1)X+1 aX X+a  X aX+1 (a+1)X+a (a+1)X+a+1  1 aX+a  X  X aX+a+1 (a+1)X+a aX+1 (a+1)X+a+1 aX+a  0 (a+1)X aX+1 aX+a+1 aX+1 aX+a+1  X (a+1)X+a (a+1)X+a aX+a  X a+1 X+a+1 a+1 aX (a+1)X+a+1 a+1 (a+1)X+a aX+a aX+a+1  1  X X+a+1  1 (a+1)X+a  a  a X+1  0  a  0  1  1  1 aX aX+a+1  1  X aX+1 (a+1)X aX+a+1  1  a
 0  0  0 (a+1)X  0  0  0  X (a+1)X  X aX (a+1)X  0  0 (a+1)X aX  X  X  0  X aX aX  0 aX (a+1)X (a+1)X  X (a+1)X aX  0 (a+1)X (a+1)X (a+1)X aX  0 aX  X  X aX aX  X aX (a+1)X  X aX aX (a+1)X (a+1)X  X (a+1)X  0 (a+1)X  0  X (a+1)X  0  0  X aX aX  0  X  0  0  X aX aX  X (a+1)X  0 aX (a+1)X (a+1)X  0  0  0  X (a+1)X  X
 0  0  0  0  X aX aX aX aX  0 (a+1)X  X  0 (a+1)X (a+1)X  X  X  0  X aX  X aX  0 (a+1)X (a+1)X (a+1)X  0 aX aX (a+1)X  X  X (a+1)X  0  X  0 (a+1)X (a+1)X (a+1)X  0 (a+1)X  X  X (a+1)X aX aX  0  X (a+1)X  0 (a+1)X (a+1)X  X aX  0 (a+1)X  X  0  0  0 aX (a+1)X  0  X  X  0 aX aX  X aX aX (a+1)X  0  0  0 aX aX (a+1)X  0

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

Homogenous weight enumerator: w(x)=1x^0+585x^220+264x^221+228x^222+672x^223+2355x^224+1008x^225+744x^226+1200x^227+3819x^228+1596x^229+864x^230+1860x^231+5727x^232+2124x^233+936x^234+2472x^235+6453x^236+2244x^237+1104x^238+2316x^239+6438x^240+2076x^241+936x^242+1776x^243+5091x^244+1956x^245+768x^246+1452x^247+2997x^248+708x^249+456x^250+504x^251+1086x^252+276x^253+108x^254+36x^255+162x^256+36x^257+24x^260+33x^264+24x^268+12x^272+6x^276+3x^280

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 23.7 seconds.