The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+200x^81+438x^82+376x^83+283x^84+512x^85+474x^86+512x^87+290x^88+376x^89+428x^90+200x^91+2x^94+1x^96+2x^114+1x^116

The gray image is a linear code over GF(2) with n=688, k=12 and d=324.
This code was found by Heurico 1.16 in 0.735 seconds.