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

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+638x^80+2048x^81+3515x^82+5612x^83+8168x^84+11082x^85+11838x^86+15182x^87+15326x^88+14828x^89+12623x^90+11270x^91+7560x^92+5118x^93+3058x^94+1676x^95+778x^96+416x^97+184x^98+74x^99+36x^100+12x^101+12x^102+10x^103+3x^104+2x^108+2x^110

The gray image is a linear code over GF(2) with n=704, k=17 and d=320.
This code was found by Heurico 1.16 in 205 seconds.