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

generates a code of length 81 over F4[X]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+192x^243+24x^244+36x^248+3x^256

The gray image is a linear code over GF(4) with n=324, k=4 and d=243.
This code was found by Heurico 1.16 in 0.281 seconds.