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

generates a code of length 72 over Z2[X]/(X^4) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+53x^66+190x^67+329x^68+418x^69+428x^70+482x^71+485x^72+436x^73+450x^74+320x^75+171x^76+118x^77+77x^78+50x^79+34x^80+16x^81+13x^82+14x^83+4x^84+4x^85+2x^86+1x^110

The gray image is a linear code over GF(2) with n=576, k=12 and d=264.
This code was found by Heurico 1.16 in 0.656 seconds.