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

generates a code of length 85 over Z2[X]/(X^4) who´s minimum homogenous weight is 87.

Homogenous weight enumerator: w(x)=1x^0+20x^87+4x^88+4x^89+2x^90+1x^92

The gray image is a linear code over GF(2) with n=680, k=5 and d=348.
This code was found by Heurico 1.16 in 0.407 seconds.