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

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+84x^86+16x^87+214x^88+48x^89+253x^90+160x^91+236x^92+336x^93+210x^94+464x^95+224x^96+480x^97+182x^98+320x^99+184x^100+144x^101+134x^102+64x^103+90x^104+16x^105+108x^106+54x^108+44x^110+15x^112+8x^114+5x^116+1x^122+1x^148

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 2.69 seconds.