The generator matrix

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

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+364x^67+500x^68+558x^69+557x^70+436x^71+404x^72+486x^73+350x^74+256x^75+84x^76+54x^77+5x^78+4x^79+12x^80+2x^81+12x^83+5x^84+4x^85+1x^92+1x^96

The gray image is a linear code over GF(2) with n=568, k=12 and d=268.
This code was found by Heurico 1.16 in 10.7 seconds.