The generator matrix

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

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+142x^78+64x^79+413x^80+264x^81+634x^82+436x^83+727x^84+472x^85+785x^86+548x^87+714x^88+584x^89+668x^90+412x^91+444x^92+200x^93+310x^94+76x^95+132x^96+16x^97+52x^98+45x^100+19x^102+12x^104+14x^106+6x^108+2x^116

The gray image is a linear code over GF(2) with n=348, k=13 and d=156.
This code was found by Heurico 1.16 in 5.91 seconds.