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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+144x^83+112x^84+180x^85+182x^86+224x^87+58x^88+40x^89+24x^90+48x^91+4x^92+4x^93+2x^102+1x^128

The gray image is a linear code over GF(2) with n=688, k=10 and d=332.
This code was found by Heurico 1.16 in 0.984 seconds.