The generator matrix

 1  0  0  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1 a^2*X  1  0  1  X  1 a*X  1  1  1  1 a^2*X  1  1  1 a*X  1  1  1  1  1  1  1  1  1  1  0  1 a^2*X  1  1  1  X  1  1  X  1  1  1  1  1  1  X  0  1 a*X  1  1 a*X  1 a^2*X  1  1 a*X  1  1  1  0  1  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1 a*X
 0  1  0 a^2*X a*X  X  1 a^2*X+a a^2 a^2*X+1 a*X+1 X+1 a*X+a  1 X+a^2 X+a a^2*X+a^2  a  1 a*X+a^2  1  a  1  0  1 a*X+1 a^2*X X+a^2 a^2  1 X+1 a^2*X+a  X  0 a*X+a^2 a^2*X+a X+a a*X+1 a^2 a^2*X a*X+a^2  1 a*X+a X+a^2  1 a^2*X+1  1 a*X+a a*X+a^2 a*X  1 a^2*X+a^2  0  1 X+a  a a^2*X+a a^2 a*X+a^2  1  1  1 X+1  1 a^2 X+a^2  1 X+a^2  1 a^2*X+1 X+1  1 a*X+a a*X+a X+a a^2*X  1  1 a*X+1 a*X+1  X X+a a*X+a^2 X+1 X+1 a^2*X+1 a*X a^2*X a*X  1  1  0 a^2*X a*X a*X  1
 0  0  1  1  a a^2 X+a^2 a^2*X+a^2 a*X+a^2 X+a a*X+1  X X+1 a^2*X+a^2 a*X+a a^2*X+a a^2*X+1  0 a*X+1 a*X X+a  X a^2*X+a a*X a^2*X+1 X+a^2 a^2*X+1 a*X+1 a*X+a  X X+a a^2*X+a a^2*X+a^2  1 X+a^2  X  1  0 a^2*X+1 a^2 a^2*X+a a*X+a X+a  a  1 a*X+1 a^2*X a^2*X+a^2  a a^2*X a*X+a a^2*X  1 a^2 a^2*X a^2 X+a a^2*X+a a^2*X a^2  0 a^2*X+1 a*X+a^2 a^2 a*X+1 a*X a*X+1 X+1 a^2*X+a^2  1 a*X+a a*X+a^2 a^2 a*X+a  0  1 a^2*X+a  X a^2*X+1 a*X+a^2 a*X+1 X+a^2 a^2*X+a^2 a^2 a*X a^2*X+a  0 a*X+a^2 X+1 a^2*X+1  1 X+1 a*X+a  1 a^2 a*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+528x^281+240x^282+492x^283+117x^284+576x^285+264x^286+300x^287+63x^288+312x^289+120x^290+228x^291+42x^292+252x^293+48x^294+36x^295+12x^296+96x^297+36x^298+24x^299+6x^300+48x^301+12x^302+24x^305+36x^306+24x^307+36x^309+12x^310+48x^311+12x^312+48x^313+3x^332

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 1.48 seconds.