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  1  1  1  1
 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 a^2  1 a^2*X+a^2 a*X X+a X+a^2 X+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 a^2*X+a^2 a^2*X+a a^2*X a^2*X a*X+a^2 a^2*X+1 a^2*X+a
 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 a^2*X a^2*X a*X  0 a*X a*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+768x^243+417x^244+2472x^247+801x^248+2496x^251+759x^252+2136x^255+786x^256+1572x^259+585x^260+1200x^263+381x^264+948x^267+219x^268+564x^271+108x^272+120x^275+33x^276+12x^279+3x^288+3x^292

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 1.33 seconds.