The generator matrix

 1  0  1  1 X^2  1  1  1 X^2+X  1  1  0 X^3+X  1  1  1  1 X^2 X^3  1  1  1  1  1 X^3  1 X^3+X  1  1  1  1 X^3+X^2+X  1 X^2+X  1  1  1 X^2+X  1 X^3+X  1  1  1  1  1 X^3+X^2+X  0 X^3+X^2+X  0  1  1  X  1  1  1  1  1  1 X^3+X^2+X X^3+X  1 X^2  1  1 X^3+X^2  1  X  0  1  1 X^2  1
 0  1  1 X^2+X  1 X^2+X+1 X^2 X^3+1  1 X+1 X^3+X^2+X  1  1  0 X^3+X^2+1 X^3 X^3+1  1  1 X^2+X X+1  X X+1 X^3+X^2  1  1  1 X^2 X^3 X^2+1 X^3+1  1 X^3+X^2+X+1  1 X^3+X X^2+X+1 X^3  1 X^3+X+1  1 X+1  X X^2+1  X X^3+X^2+X  1  1  1  1 X^3+X^2+X+1 X^3+X^2+X X^3+X^2+X  1  X X^2  X X^2+1 X^3+X^2+1  1  1  X  1 X^3+1 X^3+1  1 X^2+X+1 X^3+X^2  X  1 X^3+X^2+X  1  1
 0  0  X  0 X^3+X  X X^3+X X^3  0 X^3 X^3+X X^3+X^2+X X^2 X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X X^2+X X^3+X X^2+X X^2 X^2 X^2+X X^2+X X^3+X^2 X^2 X^3+X^2+X X^2 X^3+X X^3  X X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X  X  0  X X^3+X^2  0 X^3+X^2+X X^3+X^2 X^3+X^2+X X^3+X X^3 X^3+X X^3+X^2 X^3  0  0 X^2 X^2+X X^3 X^2+X X^3+X  X X^3+X X^2 X^3+X^2+X X^3+X^2  0  X X^3+X^2 X^2+X X^2 X^2+X X^3+X^2+X X^3+X^2+X X^3+X^2  X  0 X^3+X^2
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3 X^3 X^3  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+382x^68+458x^69+567x^70+512x^71+586x^72+424x^73+412x^74+300x^75+204x^76+70x^77+105x^78+20x^79+22x^80+8x^81+18x^82+4x^84+1x^88+2x^94

The gray image is a linear code over GF(2) with n=576, k=12 and d=272.
This code was found by Heurico 1.16 in 0.672 seconds.