The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+221x^80+435x^82+32x^83+712x^84+224x^85+924x^86+224x^87+715x^88+32x^89+344x^90+145x^92+48x^94+15x^96+9x^98+14x^100+1x^148

The gray image is a linear code over GF(2) with n=688, k=12 and d=320.
This code was found by Heurico 1.16 in 1.47 seconds.