The generator matrix

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

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+261x^66+168x^67+569x^68+232x^69+767x^70+240x^71+701x^72+240x^73+510x^74+104x^75+169x^76+40x^77+61x^78+30x^80+2x^84+1x^122

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 22.7 seconds.