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

generates a code of length 78 over F4[X]/(X^2) who´s minimum homogenous weight is 228.

Homogenous weight enumerator: w(x)=1x^0+66x^228+99x^232+768x^234+36x^236+3x^240+42x^244+6x^248+3x^312

The gray image is a linear code over GF(4) with n=312, k=5 and d=228.
This code was found by Heurico 1.16 in 0.156 seconds.