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

generates a code of length 96 over Z2[X]/(X^3) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+37x^84+22x^85+55x^86+50x^87+114x^88+132x^89+149x^90+184x^91+225x^92+278x^93+290x^94+356x^95+363x^96+368x^97+340x^98+288x^99+180x^100+170x^101+91x^102+122x^103+58x^104+44x^105+60x^106+24x^107+30x^108+10x^109+21x^110+8x^112+8x^114+7x^116+6x^118+2x^122+1x^124+1x^126+1x^154

The gray image is a linear code over GF(2) with n=384, k=12 and d=168.
This code was found by Heurico 1.16 in 2.95 seconds.