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

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+136x^67+115x^68+284x^69+289x^70+514x^71+271x^72+216x^73+43x^74+60x^75+45x^76+60x^77+3x^78+10x^79+1x^130

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 0.438 seconds.