The generator matrix

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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+176x^80+744x^81+1580x^82+2400x^83+2874x^84+3612x^85+3667x^86+3786x^87+3487x^88+3356x^89+2481x^90+1796x^91+1064x^92+772x^93+471x^94+198x^95+138x^96+60x^97+43x^98+24x^99+12x^100+16x^101+6x^102+4x^103

The gray image is a linear code over GF(2) with n=696, k=15 and d=320.
This code was found by Heurico 1.16 in 14.8 seconds.