The generator matrix

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

generates a code of length 86 over F4[X]/(X^2) who´s minimum homogenous weight is 257.

Homogenous weight enumerator: w(x)=1x^0+72x^257+144x^258+24x^261+3x^264+3x^268+6x^280+3x^284

The gray image is a linear code over GF(4) with n=344, k=4 and d=257.
This code was found by Heurico 1.16 in 0.093 seconds.