The generator matrix

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

generates a code of length 72 over Z2[X]/(X^4) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+246x^65+1374x^66+2572x^67+4178x^68+5800x^69+6747x^70+8310x^71+7888x^72+8176x^73+6789x^74+5256x^75+3686x^76+2228x^77+1341x^78+554x^79+191x^80+106x^81+53x^82+24x^83+8x^84+4x^85+4x^87

The gray image is a linear code over GF(2) with n=576, k=16 and d=260.
This code was found by Heurico 1.16 in 40 seconds.