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

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

Homogenous weight enumerator: w(x)=1x^0+180x^234+480x^235+411x^236+372x^237+696x^238+2052x^239+1326x^240+744x^241+1680x^242+3048x^243+1815x^244+1428x^245+2388x^246+4428x^247+2076x^248+1464x^249+2688x^250+4800x^251+2154x^252+1860x^253+2748x^254+5064x^255+2193x^256+1620x^257+2508x^258+4272x^259+1776x^260+1164x^261+1596x^262+2364x^263+1059x^264+480x^265+708x^266+912x^267+324x^268+72x^269+156x^270+204x^271+90x^272+12x^273+12x^274+24x^275+21x^276+18x^280+12x^284+15x^288+12x^292+3x^300+3x^304+3x^312

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 25.8 seconds.