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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 85.

Homogenous weight enumerator: w(x)=1x^0+28x^85+205x^86+13x^88+2x^89+2x^92+2x^97+2x^98+1x^110

The gray image is a linear code over GF(2) with n=688, k=8 and d=340.
This code was found by Heurico 1.16 in 29.3 seconds.