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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+362x^66+48x^67+509x^68+304x^69+644x^70+592x^71+505x^72+272x^73+438x^74+64x^75+219x^76+80x^78+45x^80+12x^82+1x^112

The gray image is a linear code over GF(2) with n=568, k=12 and d=264.
This code was found by Heurico 1.16 in 5.97 seconds.