The generator matrix

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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+83x^78+112x^79+304x^80+152x^81+394x^82+268x^83+352x^84+208x^85+392x^86+292x^87+370x^88+216x^89+308x^90+212x^91+204x^92+64x^93+71x^94+12x^95+27x^96+18x^98+11x^100+14x^102+8x^104+2x^112+1x^116

The gray image is a linear code over GF(2) with n=344, k=12 and d=156.
This code was found by Heurico 1.16 in 1.63 seconds.