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  X  X  1  1  X  X  1  1  1  1  X  X  X  X  X  X  X  X  X  X X^2  0 X^2  0  1  1  1  1  1  1  1  1 X^2  0 X^2  0  X  X X^2  2 X^2  2 X^2  2 X^2  2  1
 0 X^2+2  0 X^2+2  0 X^2+2  0 X^2+2  2 X^2  2 X^2  2 X^2  2 X^2  0 X^2+2  0 X^2+2  0 X^2+2  0 X^2+2  2 X^2 X^2+2 X^2+2  2 X^2 X^2+2 X^2+2  2 X^2  2 X^2  0  2 X^2  0  2 X^2  0  2 X^2 X^2 X^2+2 X^2 X^2+2 X^2  0  2  0  2  0  2  0  2 X^2+2 X^2 X^2+2 X^2  0  2 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0
 0  0  2  0  0  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  2  2  2  2  2  2  0  2  0  0  0  2  2  2  0  0  0  0  2  2  2  0  2  2  2  0  0  0  2  2  0  0  0  0  2  2  2  2  0  0  2  2  0  0  2  2  0  0  2  2  0  0  0
 0  0  0  2  2  2  2  0  0  0  2  2  2  2  0  0  0  0  2  2  2  2  0  0  0  0  2  2  0  0  0  0  2  2  2  2  2  2  0  2  2  0  0  0  2  2  0  2  2  0  0  0  2  2  2  2  0  0  2  0  0  2  0  0  0  2  2  0  2  0  0  2  0

generates a code of length 73 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+14x^72+222x^73+14x^74+1x^80+1x^82+2x^89+1x^98

The gray image is a code over GF(2) with n=584, k=8 and d=288.
This code was found by Heurico 1.16 in 0.359 seconds.