The generator matrix

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

generates a code of length 81 over F4[X]/(X^2) who´s minimum homogenous weight is 224.

Homogenous weight enumerator: w(x)=1x^0+162x^224+120x^225+444x^227+1566x^228+576x^229+864x^231+3804x^232+1224x^233+1728x^235+5670x^236+1980x^237+2148x^239+7011x^240+2100x^241+2028x^243+7830x^244+2424x^245+1992x^247+7182x^248+2028x^249+1752x^251+4902x^252+1164x^253+900x^255+2019x^256+564x^257+384x^259+678x^260+96x^261+48x^263+60x^264+12x^265+21x^268+15x^272+12x^276+12x^280+9x^284+3x^288+3x^304

The gray image is a linear code over GF(4) with n=324, k=8 and d=224.
This code was found by Heurico 1.16 in 24.6 seconds.