The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+513x^82+448x^83+580x^84+412x^85+468x^86+364x^87+408x^88+340x^89+337x^90+100x^91+68x^92+30x^94+8x^96+12x^98+4x^100+1x^108+1x^112+1x^116

The gray image is a linear code over GF(2) with n=688, k=12 and d=328.
This code was found by Heurico 1.16 in 34.5 seconds.