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 X^3  1  1  1  1  0  X  X  0  X  X
 0  X X^3+X^2 X^2+X  0 X^2+X X^3+X^2 X^3+X X^2+X  0 X^3+X X^3+X^2 X^3+X X^3 X^2 X^3+X^2+X X^3+X  0 X^2+X X^3+X^2 X^2+X  0 X^3+X^2 X^3+X X^3+X^2+X  0 X^3+X X^3+X^2 X^3+X^2+X X^3  X X^2 X^2+X  0  0  X X^3+X^2 X^3+X^2 X^3+X^2+X  X X^3 X^2 X^2+X X^2+X X^2 X^3 X^3+X X^3+X  0 X^3  0 X^3 X^2+X X^3+X^2+X X^2+X X^3+X^2+X X^3+X^2 X^3+X^2 X^2 X^2 X^3 X^3+X X^3+X  X  X  X X^2+X X^3+X  X  0  0
 0  0 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3  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  0  0  0 X^3 X^3  0  0 X^3 X^3  0  0  0  0  0 X^3  0
 0  0  0 X^3  0  0  0  0  0  0  0  0  0  0  0 X^3  0 X^3  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  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3  0  0  0 X^3 X^3  0
 0  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 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  0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0
 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  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0  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+44x^66+318x^68+256x^69+328x^70+256x^71+510x^72+108x^74+194x^76+32x^78+1x^128

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 0.437 seconds.