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  X  1  1  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X X^2+X  0 X^2+X X^2 X^2  X X^3 X^3+X  0 X^2+X X^2  X X^3 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^3 X^3+X^2+X X^2  X X^2 X^2+X X^3+X^2 X^3+X  0 X^3+X^2+X  0  X  0 X^2+X X^3+X^2 X^2+X  0 X^3+X X^3+X^2 X^3+X X^3+X^2 X^3+X  0 X^3 X^2+X X^3 X^3+X^2+X X^3+X^2+X X^2+X X^3+X^2 X^3+X^2 X^3  0 X^3+X X^3+X X^3+X^2 X^3+X X^2  X X^3+X X^3+X X^2+X X^3 X^3+X^2+X  0
 0  0 X^3+X^2  0 X^2 X^2  0 X^2  0  0 X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3 X^3 X^2 X^2 X^2 X^3+X^2 X^3  0 X^3 X^3 X^2 X^2  0 X^3+X^2 X^2  0 X^3+X^2 X^2 X^3 X^2 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3 X^3  0 X^3+X^2 X^3+X^2  0  0  0 X^2  0  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0  0 X^2 X^3 X^3+X^2 X^2 X^2 X^3  0  0 X^3  0 X^3 X^2  0
 0  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0  0  0 X^3  0  0  0 X^3  0  0
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0 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+48x^66+52x^67+127x^68+256x^69+476x^70+248x^71+567x^72+32x^73+55x^74+20x^75+89x^76+32x^77+44x^78+1x^138

The gray image is a linear code over GF(2) with n=568, k=11 and d=264.
This code was found by Heurico 1.16 in 0.594 seconds.