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

generates a code of length 87 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+127x^78+40x^79+269x^80+52x^81+414x^82+124x^83+385x^84+180x^85+467x^86+208x^87+432x^88+208x^89+329x^90+124x^91+200x^92+68x^93+158x^94+16x^95+119x^96+4x^97+77x^98+50x^100+28x^102+15x^104+1x^124

The gray image is a linear code over GF(2) with n=348, k=12 and d=156.
This code was found by Heurico 1.16 in 2.12 seconds.