The generator matrix

 1  0  0  1  1  1 X^3  1  1  1 X^3+X^2+X X^2+X  1 X^2+X  1 X^3+X^2 X^3+X  X  1  0  1  1  1  1 X^3+X X^2+X  1  1  1  1 X^3 X^2+X X^2+X  1  1  1  1 X^2  1  1 X^3+X^2+X  X  1  0  1 X^3+X  1 X^2  1  1  X  1  X X^3+X^2 X^3+X^2  1  1  1 X^3  1  1 X^2+X  1  1  1  1 X^3+X^2  1  1  0  1
 0  1  0 X^2 X^3+X^2+1 X^2+1  1 X^3+X^2 X^3 X+1  1  0 X^3+X+1  1 X^3+X^2+X  1  1 X^2+X X^3+X^2+X+1  1 X^2+X+1 X^3+X X^2+X X^3+X^2+1  1 X^3+X^2 X^3+X^2 X^3+X+1 X^3+X  X  X  1  1 X^2 X^3+X^2+1 X^3+1  1  1 X^3+X^2+X X+1 X^2+X  1 X^3+1 X^3 X^3 X^3+X X^3+X+1  1  1 X^3  1 X^2+1  1  1 X^2+X X^2+X+1 X^3+X+1 X^3+1  1 X^2+1  X  1 X^3+X^2+X+1 X^3 X^2 X^3+1  X  1 X^2+X+1  1 X^3
 0  0  1 X^2+X+1 X^3+X^2+X+1 X^3+X^2 X^3+X+1 X^3+X X^3+X^2+1 X^3 X+1  1  1  0 X^3+X^2+X X^2+1 X^2+X  1 X+1 X^3+X X^2+X X^2+1 X^2 X^3+X X^3+X^2+1  1 X+1 X^3+X+1 X^2+X+1 X^3+X^2+X  1 X^3+X^2+X X^2 X^3+1 X^2+1 X^3+X^2+X+1 X^3+X^2  1  0 X^2+X  1 X^3+X+1 X^2+X  1  0  1 X^3+1 X^2+X X^3+X X^3+X^2+X X^3+X^2+1 X^3 X^3+X^2+X+1 X+1  1 X^3+X X^3 X^3+X+1 X^2 X^2+X X+1 X^2 X^2+X+1 X^3+X X^3+X^2  1  1 X^3+X^2+1 X^2+X+1 X^2 X^3+X^2
 0  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3  0  0  0 X^3  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+79x^66+530x^67+981x^68+1286x^69+1175x^70+1180x^71+829x^72+632x^73+407x^74+358x^75+287x^76+222x^77+113x^78+68x^79+28x^80+12x^81+2x^84+2x^86

The gray image is a linear code over GF(2) with n=568, k=13 and d=264.
This code was found by Heurico 1.16 in 2.2 seconds.