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

generates a code of length 90 over F4[X]/(X^2) who´s minimum homogenous weight is 270.

Homogenous weight enumerator: w(x)=1x^0+192x^270+45x^272+18x^280

The gray image is a linear code over GF(4) with n=360, k=4 and d=270.
This code was found by Heurico 1.16 in 0.125 seconds.