The generator matrix

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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+530x^78+1580x^79+3892x^80+6528x^81+10666x^82+15620x^83+21273x^84+24436x^85+31690x^86+29800x^87+30616x^88+26540x^89+21606x^90+14756x^91+10741x^92+5760x^93+3160x^94+1572x^95+732x^96+348x^97+202x^98+32x^99+32x^100+4x^101+16x^102+7x^104+2x^106+2x^116

The gray image is a linear code over GF(2) with n=696, k=18 and d=312.
This code was found by Heurico 1.16 in 823 seconds.