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

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+67x^80+240x^81+287x^82+338x^83+412x^84+560x^85+421x^86+536x^87+429x^88+256x^89+203x^90+182x^91+54x^92+28x^93+23x^94+24x^95+20x^96+4x^97+2x^98+8x^99+1x^144

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 1.09 seconds.