The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  1  0  1 X^3+X^2+X  1  1  1  1 X^3  1 X^3+X  1  1  0  1  1 X^3+X  1  1  1 X^2+X  1 X^2  1  1  1 X^2  1  1 X^2  1 X^3+X^2+X X^3+X^2+X  1  1  1  1  X  1  1 X^3+X^2  1 X^3+X X^3+X^2  1  0  1  1  1  1  1  X  1  1 X^3+X^2  1 X^2+X X^3  1  1  1  0  1  1
 0  1 X+1 X^3+X^2+X X^2+1  1 X^3+X^2+1  0  1 X^3+X^2+X  1 X+1 X^3+1 X^2+X+1 X^3  1 X^3+X  1 X^3+X^2+X+1  0  1  1 X^3+X  1 X^3+X^2+X+1 X^3+X^2+1 X^2  1 X^2+X  1 X+1 X^3+X^2+X X^2+1  1 X^3+X^2+X+1 X^3  1 X^3+X^2+1  1  1 X^3+X^2+X+1  1 X^2+1 X^2 X^2 X+1 X^3+X+1  1 X^3+X  1  X X^2+1  1 X^3+X^2+X+1  X X^3+1 X^3+X^2+X X^2+X  1  0  X  1 X^3+X^2+1  1  1 X+1 X^3+X X^3+X^2+X  1 X^3+X^2  0
 0  0 X^2  0  0  0  0 X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^2 X^3+X^2 X^2 X^3 X^3 X^2 X^3 X^3 X^3 X^2 X^3+X^2 X^3+X^2 X^2  0 X^2 X^2 X^3  0  0 X^3+X^2 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^2 X^3 X^3 X^3 X^3 X^3 X^2 X^3+X^2 X^3 X^3+X^2 X^3  0 X^3 X^3+X^2 X^2 X^3 X^2  0 X^3+X^2  0 X^2  0 X^3 X^2 X^3  0 X^3 X^2 X^3 X^3
 0  0  0 X^3+X^2 X^3 X^3+X^2 X^2 X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^3  0 X^3 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^2  0 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3  0 X^3+X^2 X^3 X^2 X^3  0 X^2 X^2 X^3  0 X^3 X^2  0 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3  0  0 X^3 X^3+X^2  0 X^2 X^2 X^2 X^3 X^3 X^3+X^2 X^3  0  0 X^3+X^2 X^2  0 X^3+X^2 X^3+X^2  0 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+78x^66+310x^67+506x^68+442x^69+561x^70+488x^71+461x^72+396x^73+447x^74+212x^75+97x^76+56x^77+16x^78+12x^79+2x^80+2x^83+2x^84+2x^85+2x^88+1x^90+1x^92+1x^94

The gray image is a linear code over GF(2) with n=568, k=12 and d=264.
This code was found by Heurico 1.16 in 0.469 seconds.