The generator matrix

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

generates a code of length 96 over F4[X]/(X^2) who´s minimum homogenous weight is 271.

Homogenous weight enumerator: w(x)=1x^0+576x^271+717x^272+624x^273+768x^274+1812x^275+1770x^276+1620x^277+1284x^278+2628x^279+2511x^280+1872x^281+1668x^282+3888x^283+2709x^284+2352x^285+1704x^286+3744x^287+2760x^288+2268x^289+1740x^290+3636x^291+2769x^292+2172x^293+1644x^294+3276x^295+2160x^296+1476x^297+1212x^298+1992x^299+1506x^300+948x^301+468x^302+1068x^303+732x^304+432x^305+216x^306+324x^307+267x^308+60x^309+36x^310+84x^311+15x^312+12x^314+12x^315+3x^316

The gray image is a linear code over GF(4) with n=384, k=8 and d=271.
This code was found by Heurico 1.16 in 34.3 seconds.