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

generates a code of length 83 over F4[X]/(X^2) who´s minimum homogenous weight is 249.

Homogenous weight enumerator: w(x)=1x^0+192x^249+48x^252+15x^256

The gray image is a linear code over GF(4) with n=332, k=4 and d=249.
This code was found by Heurico 1.16 in 0.156 seconds.