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

generates a code of length 57 over F4[X]/(X^2) who´s minimum homogenous weight is 169.

Homogenous weight enumerator: w(x)=1x^0+132x^169+45x^172+48x^173+15x^176+12x^185+3x^188

The gray image is a linear code over GF(4) with n=228, k=4 and d=169.
This code was found by Heurico 1.16 in 0.046 seconds.