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

generates a code of length 84 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 234.

Homogenous weight enumerator: w(x)=1x^0+180x^234+336x^235+795x^236+252x^237+1116x^238+1116x^239+1893x^240+528x^241+1872x^242+2340x^243+2883x^244+912x^245+2844x^246+2736x^247+3534x^248+1104x^249+3084x^250+3552x^251+4188x^252+1128x^253+3204x^254+3156x^255+3837x^256+1008x^257+3012x^258+2820x^259+3069x^260+720x^261+2052x^262+1704x^263+1515x^264+432x^265+864x^266+552x^267+618x^268+60x^269+192x^270+120x^271+105x^272+12x^274+24x^276+12x^280+27x^284+12x^288+12x^292+3x^296

The gray image is a linear code over GF(4) with n=336, k=8 and d=234.
This code was found by Heurico 1.16 in 45.6 seconds.