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

generates a code of length 87 over F4[X]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+216x^243+483x^244+336x^245+564x^246+1464x^247+1293x^248+864x^249+912x^250+2916x^251+2547x^252+1212x^253+1116x^254+4524x^255+2985x^256+1284x^257+1380x^258+4380x^259+3321x^260+1692x^261+1668x^262+5100x^263+3189x^264+1608x^265+1716x^266+4620x^267+2925x^268+1164x^269+1140x^270+2868x^271+1728x^272+660x^273+588x^274+1308x^275+636x^276+276x^277+120x^278+252x^279+237x^280+96x^281+12x^282+48x^284+24x^285+12x^288+12x^292+9x^296+12x^300+18x^304

The gray image is a linear code over GF(4) with n=348, k=8 and d=243.
This code was found by Heurico 1.16 in 27 seconds.