The generator matrix

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

generates a code of length 96 over F4[X]/(X^2) who´s minimum homogenous weight is 280.

Homogenous weight enumerator: w(x)=1x^0+60x^280+54x^284+192x^285+69x^288+576x^289+21x^292+18x^296+15x^300+6x^304+3x^308+6x^312+3x^380

The gray image is a linear code over GF(4) with n=384, k=5 and d=280.
This code was found by Heurico 1.16 in 0.172 seconds.