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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+80x^67+215x^68+288x^69+336x^70+256x^71+598x^72+32x^73+48x^75+129x^76+64x^77+1x^136

The gray image is a linear code over GF(2) with n=568, k=11 and d=268.
This code was found by Heurico 1.16 in 0.422 seconds.