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  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  X  1  X  1  X  X X^2  1
 0 X^3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 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  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0  0 X^3 X^3
 0  0 X^3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0  0  0 X^3 X^3 X^3
 0  0  0 X^3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3
 0  0  0  0 X^3  0  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 X^3 X^3  0 X^3  0 X^3  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  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0  0 X^3  0  0 X^3  0 X^3  0  0 X^3
 0  0  0  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3
 0  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3  0 X^3  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  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0  0  0  0  0  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3  0 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+112x^80+48x^82+256x^85+1184x^86+256x^87+94x^88+48x^90+46x^96+2x^104+1x^160

The gray image is a linear code over GF(2) with n=688, k=11 and d=320.
This code was found by Heurico 1.16 in 7.97 seconds.