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

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+143x^72+156x^73+277x^74+312x^75+422x^76+232x^77+285x^78+32x^79+62x^80+28x^81+73x^82+8x^83+12x^84+4x^86+1x^142

The gray image is a linear code over GF(2) with n=608, k=11 and d=288.
This code was found by Heurico 1.16 in 0.516 seconds.