The generator matrix

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

generates a code of length 96 over F4[X]/(X^2) who´s minimum homogenous weight is 281.

Homogenous weight enumerator: w(x)=1x^0+852x^281+711x^284+1020x^285+60x^288+468x^289+420x^297+246x^300+132x^301+180x^305+3x^316+3x^320

The gray image is a linear code over GF(4) with n=384, k=6 and d=281.
This code was found by Heurico 1.16 in 15.9 seconds.