The generator matrix

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

generates a code of length 86 over F4[X]/(X^2) who´s minimum homogenous weight is 256.

Homogenous weight enumerator: w(x)=1x^0+183x^256+768x^258+60x^264+12x^280

The gray image is a linear code over GF(4) with n=344, k=5 and d=256.
This code was found by Heurico 1.16 in 0.094 seconds.