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

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+552x^271+630x^272+804x^273+720x^274+1968x^275+1668x^276+1500x^277+1368x^278+2880x^279+2262x^280+1944x^281+1668x^282+3504x^283+2949x^284+2304x^285+1704x^286+3480x^287+3105x^288+2148x^289+1656x^290+3828x^291+2832x^292+1968x^293+1716x^294+3168x^295+2304x^296+1656x^297+1140x^298+2196x^299+1317x^300+972x^301+528x^302+1020x^303+639x^304+396x^305+192x^306+360x^307+162x^308+120x^309+60x^310+84x^311+48x^312+12x^313+3x^320

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 34.1 seconds.