The generator matrix

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

generates a code of length 80 over F4[X]/(X^2) who�s minimum homogenous weight is 224.

Homogenous weight enumerator: w(x)=1x^0+558x^224+900x^225+804x^226+192x^227+1668x^228+2568x^229+1776x^230+384x^231+2898x^232+4536x^233+2064x^234+516x^235+3549x^236+4956x^237+2160x^238+648x^239+3765x^240+4596x^241+2472x^242+552x^243+3333x^244+4332x^245+2148x^246+408x^247+3024x^248+3624x^249+1608x^250+252x^251+1611x^252+1740x^253+624x^254+96x^255+498x^256+360x^257+156x^258+24x^259+87x^260+36x^261+12x^262

The gray image is a linear code over GF(4) with n=320, k=8 and d=224.
This code was found by Heurico 1.16 in 28 seconds.