The generator matrix

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

generates a code of length 80 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+297x^72+1614x^73+3090x^74+5170x^75+8350x^76+10990x^77+13080x^78+14838x^79+16323x^80+15174x^81+13942x^82+10404x^83+7491x^84+5042x^85+2753x^86+1464x^87+571x^88+270x^89+114x^90+42x^91+21x^92+12x^93+12x^94+2x^95+2x^96+2x^97+1x^110

The gray image is a linear code over GF(2) with n=640, k=17 and d=288.
This code was found by Heurico 1.16 in 187 seconds.