The generator matrix

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

generates a code of length 85 over F4[X]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+816x^243+456x^244+2208x^247+930x^248+2688x^251+693x^252+2100x^255+663x^256+1644x^259+477x^260+1284x^263+360x^264+996x^267+330x^268+408x^271+171x^272+144x^275+9x^276+3x^284+3x^288

The gray image is a linear code over GF(4) with n=340, k=7 and d=243.
This code was found by Heurico 1.16 in 1.66 seconds.