The generator matrix

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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+532x^79+1760x^80+3782x^81+5718x^82+7788x^83+11107x^84+12784x^85+14948x^86+14928x^87+14964x^88+12896x^89+10801x^90+7704x^91+5096x^92+3114x^93+1695x^94+790x^95+361x^96+156x^97+77x^98+32x^99+21x^100+7x^102+2x^103+2x^104+2x^105+2x^109+2x^110

The gray image is a linear code over GF(2) with n=696, k=17 and d=316.
This code was found by Heurico 1.16 in 206 seconds.