The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  1 X^3  1  1 X^2+X X^2 X^3+X  1  1  1  1  X  1  1 X^3+X^2  1  1 X^3+X X^3  1  1 X^3+X^2+X  1  1 X^3+X  1  1 X^2  1  1 X^2  1  0 X^3+X^2+X  1  1  1 X^3+X^2+X  1  1  0  1  1  0  X X^2+X  X X^3 X^3+X  X X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2+X  X X^3+X^2+X  0 X^2+X X^2+X  1  1  1  1  1  1  X  1  1  1 X^2+X  1  X  1 X^3+X^2  1  0 X^2+X  1
 0  1 X+1 X^3+X^2+X X^2+1  1 X^3+X^2+1  0  1 X^3+X^2+X X+1  1  1  1 X^3 X^2+1 X^3+X X^3+X+1  1 X+1  0  1 X^3+X X^2+1  1  1 X^2 X^3+X^2+X+1  1  1  X  1 X^2+X+1 X^2+X  1 X^3+X^2  1  1 X^2  1  1 X^3+1 X^3+X X^2+X+1  1 X^2 X^2+X+1  1  1 X^3+X^2+X  1  1  1  1  1  1 X^2  1  1  1  1  1 X^3  1  1  1  1 X+1  0 X^2+1 X^3+X^2+X  0  1 X^3+X^2+X  X X+1 X^3+X^2+1  1  X X^3 X^3+X^2+X+1  1 X^3  1  1 X^3
 0  0 X^2  0  0  0  0 X^3+X^2 X^2 X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^2 X^3 X^3 X^3 X^3 X^3+X^2 X^3 X^2 X^3+X^2 X^3+X^2  0 X^3+X^2 X^2 X^3+X^2 X^2 X^3 X^3 X^3  0 X^2 X^3+X^2  0 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2  0  0 X^3 X^2  0 X^2 X^2  0 X^3 X^3+X^2 X^3  0 X^2 X^3+X^2  0 X^3 X^2 X^2  0 X^3+X^2 X^3  0  0 X^3  0 X^3  0 X^2 X^2 X^2 X^3  0 X^3+X^2 X^2 X^3+X^2  0  0 X^3 X^3+X^2 X^3+X^2 X^3 X^2
 0  0  0 X^3+X^2 X^3 X^3+X^2 X^2 X^2 X^3 X^3 X^3+X^2 X^3+X^2  0 X^2 X^2 X^2 X^3 X^3+X^2 X^3+X^2  0  0 X^3 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^2 X^3 X^3 X^3+X^2 X^3 X^2  0 X^3+X^2 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2  0  0  0 X^2  0 X^2 X^3+X^2 X^3+X^2  0  0 X^3+X^2  0 X^3 X^3+X^2  0 X^3 X^3 X^3+X^2 X^2  0 X^2 X^2 X^2 X^2 X^3 X^3 X^3 X^3 X^3  0  0 X^2 X^2 X^2 X^2 X^3  0  0 X^3+X^2  0 X^3+X^2  0 X^2 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+118x^81+384x^82+458x^83+542x^84+390x^85+549x^86+326x^87+382x^88+362x^89+362x^90+104x^91+68x^92+24x^93+6x^94+6x^95+6x^96+2x^101+2x^106+2x^107+1x^118+1x^120

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 0.797 seconds.