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

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

Homogenous weight enumerator: w(x)=1x^0+113x^72+12x^76+1x^80+1x^88

The gray image is a code over GF(2) with n=576, k=7 and d=288.
This code was found by Heurico 1.16 in 0.344 seconds.