The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+412x^77+1718x^78+3816x^79+6730x^80+10006x^81+15600x^82+20490x^83+26941x^84+29212x^85+31804x^86+29396x^87+27621x^88+21004x^89+16056x^90+9690x^91+5759x^92+2994x^93+1476x^94+780x^95+327x^96+170x^97+92x^98+16x^99+12x^100+10x^101+6x^102+4x^103+1x^112

The gray image is a linear code over GF(2) with n=688, k=18 and d=308.
This code was found by Heurico 1.16 in 809 seconds.