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  1  1  1  1  X  1  1 X^2  1  1 X^2  1  1  1
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0  0  0 X^3 X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3  0 X^3 X^3 X^2  0 X^3 X^2  0 X^2 X^3+X^2 X^3 X^2 X^2 X^2  0  0 X^3  0 X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^3 X^2  0 X^3+X^2  0 X^3  0 X^3 X^3+X^2 X^3  0 X^2 X^3 X^3+X^2 X^3+X^2 X^2  0 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^3+X^2  0 X^3+X^2 X^3+X^2 X^2
 0  0 X^3+X^2  0 X^2 X^2 X^3+X^2  0 X^2  0  0 X^2 X^2 X^2 X^3 X^3 X^3+X^2 X^3+X^2  0  0 X^3+X^2 X^3  0 X^3+X^2 X^3 X^3+X^2 X^3 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3+X^2  0  0 X^2 X^3 X^3+X^2 X^3 X^3+X^2  0 X^2 X^3+X^2 X^2 X^3  0 X^3+X^2  0  0 X^3 X^3+X^2 X^2 X^3 X^3 X^3 X^2 X^2 X^3 X^2  0  0 X^3 X^2 X^3+X^2 X^2  0 X^3+X^2 X^2 X^3+X^2 X^3 X^3+X^2
 0  0  0 X^3+X^2 X^2  0 X^3+X^2 X^2 X^2  0 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^3+X^2 X^3+X^2 X^3 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3 X^3+X^2 X^2 X^3  0  0 X^2 X^2  0 X^3+X^2 X^3  0 X^3+X^2 X^3 X^3 X^2  0 X^2  0 X^3+X^2 X^3 X^2 X^2 X^2  0 X^2 X^3+X^2 X^3+X^2 X^3 X^2 X^3 X^3 X^3 X^3 X^3  0 X^3 X^2 X^2 X^3 X^3+X^2 X^2  0  0  0
 0  0  0  0 X^3  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 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0  0  0  0  0  0  0 X^3  0 X^3  0  0 X^3  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+137x^66+51x^68+160x^69+338x^70+704x^71+310x^72+160x^73+102x^74+21x^76+58x^78+5x^82+1x^136

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 56.8 seconds.