The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+171x^224+144x^225+36x^226+408x^227+1140x^228+1080x^229+216x^230+1272x^231+2547x^232+2184x^233+300x^234+1968x^235+3084x^236+3216x^237+612x^238+2448x^239+4449x^240+4044x^241+528x^242+2748x^243+4914x^244+3924x^245+492x^246+2964x^247+4398x^248+3648x^249+492x^250+2232x^251+2751x^252+2328x^253+300x^254+936x^255+1497x^256+732x^257+84x^258+324x^259+492x^260+204x^261+12x^262+60x^263+90x^264+18x^268+9x^272+21x^276+3x^280+9x^284+3x^288+3x^292

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 26.1 seconds.