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 X^3  1  1  1  1  1  1  1  1  1  1  1  1  X  1
 0  X  0 X^3+X^2+X X^3 X^2+X  0  X  0 X^3+X^2+X  X  0 X^3+X^2+X X^3 X^3  X  0 X^3+X^2+X X^3 X^3+X  0  X X^3 X^2+X  0 X^2+X X^3 X^3+X^2+X X^3  X  0 X^2+X X^2+X X^2 X^3+X^2 X^3+X^2+X X^2  X X^2 X^3+X X^2 X^2+X X^2 X^3+X^2 X^3+X  X X^2 X^2+X X^2  X X^3+X X^2 X^2 X^3+X^2 X^3+X^2 X^2+X X^3+X X^3 X^3+X^2+X X^3+X^2 X^3+X^2 X^2  0  X  X X^2+X X^3+X^2+X X^2+X X^3+X X^3+X^2+X X^3+X^2+X X^3+X
 0  0 X^3+X^2  0  0 X^3+X^2 X^2 X^2  0  0  0  0 X^3+X^2 X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^2 X^3 X^3 X^3+X^2 X^3 X^2 X^3+X^2 X^3 X^3 X^2 X^3 X^3+X^2 X^2  0 X^3 X^2  0 X^2  0 X^2  0 X^3 X^3+X^2 X^2 X^3+X^2 X^3 X^2  0 X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^2  0 X^3 X^3+X^2  0  0  0 X^3+X^2 X^2 X^3 X^3 X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^3+X^2
 0  0  0 X^3+X^2 X^2 X^3+X^2 X^2  0 X^3 X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^3 X^3 X^2  0  0 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^3  0  0  0  0  0 X^3 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^2 X^2 X^2 X^3 X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^2  0 X^3 X^3+X^2  0  0 X^3 X^3+X^2 X^3 X^3+X^2  0 X^2 X^3 X^3 X^3+X^2 X^3 X^3+X^2 X^2  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+94x^68+100x^69+340x^70+408x^71+301x^72+520x^73+40x^74+24x^75+59x^76+100x^77+60x^78+1x^140

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 0.515 seconds.