The generator matrix

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

generates a code of length 85 over Z2[X]/(X^4) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+532x^77+1867x^78+3422x^79+5578x^80+8368x^81+10608x^82+12742x^83+14278x^84+15534x^85+15308x^86+13518x^87+10440x^88+7858x^89+5457x^90+2886x^91+1385x^92+734x^93+247x^94+164x^95+81x^96+26x^97+17x^98+4x^99+11x^100+2x^104+4x^105

The gray image is a linear code over GF(2) with n=680, k=17 and d=308.
This code was found by Heurico 1.16 in 201 seconds.