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

generates a code of length 96 over Z2[X]/(X^3) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+46x^86+58x^87+100x^88+190x^89+133x^90+346x^91+132x^92+518x^93+103x^94+472x^95+107x^96+556x^97+83x^98+406x^99+58x^100+258x^101+66x^102+122x^103+74x^104+48x^105+57x^106+50x^107+30x^108+24x^109+17x^110+12x^111+6x^112+6x^113+6x^114+6x^115+4x^116+1x^146

The gray image is a linear code over GF(2) with n=384, k=12 and d=172.
This code was found by Heurico 1.16 in 2.52 seconds.