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

generates a code of length 96 over Z2[X]/(X^3) who´s minimum homogenous weight is 85.

Homogenous weight enumerator: w(x)=1x^0+86x^85+130x^86+188x^87+272x^88+378x^89+333x^90+336x^91+468x^92+514x^93+594x^94+584x^95+645x^96+632x^97+552x^98+460x^99+408x^100+386x^101+296x^102+216x^103+155x^104+116x^105+109x^106+92x^107+84x^108+54x^109+26x^110+36x^111+14x^112+8x^113+6x^114+8x^115+2x^118+2x^121+1x^136

The gray image is a linear code over GF(2) with n=384, k=13 and d=170.
This code was found by Heurico 1.16 in 12.4 seconds.