The generator matrix

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

generates a code of length 86 over F4[X]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+312x^243+165x^244+180x^245+180x^246+1020x^247+198x^248+312x^249+396x^250+1272x^251+150x^252+612x^253+540x^254+1596x^255+129x^256+540x^257+648x^258+1500x^259+87x^260+588x^261+828x^262+1680x^263+66x^264+576x^265+396x^266+1200x^267+42x^268+252x^269+84x^270+492x^271+42x^272+12x^273+132x^275+27x^276+12x^279+24x^280+21x^284+27x^288+27x^292+6x^296+3x^300+3x^304+6x^316

The gray image is a linear code over GF(4) with n=344, k=7 and d=243.
This code was found by Heurico 1.16 in 80 seconds.