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 X^3  1  1  1  1  1  1  1  1  1  1  X  X  X
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X X^2 X^2+X X^3 X^3+X  0 X^2+X X^2  X  0 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^2  X X^3 X^3+X^2+X X^2  X X^2  X X^3 X^3+X^2+X  0 X^2+X  0 X^2+X X^2  X X^3+X^2 X^3+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^3 X^3+X^2+X X^2  X X^3+X X^2 X^3+X^2 X^3+X X^3+X^2  X X^2  X  0  0 X^3 X^2+X X^2+X X^3 X^3 X^3+X^2+X X^3+X^2+X  0  0 X^2+X X^2+X X^2+X X^2+X X^2+X
 0  0 X^3+X^2  0 X^2 X^2  0 X^2 X^3+X^2  0 X^2  0  0 X^3+X^2  0 X^3+X^2 X^3 X^3 X^3 X^3 X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^3 X^3 X^3 X^2 X^2 X^3+X^2 X^3+X^2  0  0 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3  0 X^3+X^2 X^2 X^2 X^2  0 X^3  0 X^3 X^3+X^2 X^2  0 X^3 X^3+X^2  0 X^3 X^2 X^3 X^3+X^2  0  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2  0  0 X^3
 0  0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3  0  0  0 X^3  0  0 X^3  0  0 X^3 X^3
 0  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0

generates a code of length 72 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+138x^68+128x^69+245x^70+480x^71+180x^72+544x^73+74x^74+32x^75+64x^76+96x^77+65x^78+1x^136

The gray image is a linear code over GF(2) with n=576, k=11 and d=272.
This code was found by Heurico 1.16 in 10.2 seconds.