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

generates a code of length 76 over F4[X]/(X^2) who´s minimum homogenous weight is 220.

Homogenous weight enumerator: w(x)=1x^0+72x^220+252x^224+336x^228+324x^232+24x^236+12x^240+3x^288

The gray image is a linear code over GF(4) with n=304, k=5 and d=220.
This code was found by Heurico 1.16 in 0.078 seconds.