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 X^2  0  0  X  X X^2 X^2 X^2 X^3 X^3 X^2 X^3 X^3
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3 X^2 X^3 X^2  0 X^3 X^2  0 X^3 X^2 X^2  0 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^2  0 X^3  0 X^3  0 X^3  0 X^3 X^3+X^2 X^3+X^2 X^2 X^2  0 X^3 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2
 0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3  0
 0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3

generates a code of length 72 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+250x^72+2x^80+2x^88+1x^96

The gray image is a linear code over GF(2) with n=576, k=8 and d=288.
This code was found by Heurico 1.16 in 0.344 seconds.