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

generates a code of length 89 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+115x^78+318x^80+28x^81+463x^82+88x^83+626x^84+264x^85+794x^86+440x^87+885x^88+448x^89+856x^90+392x^91+749x^92+248x^93+484x^94+104x^95+284x^96+36x^97+242x^98+161x^100+97x^102+43x^104+19x^106+4x^108+2x^110+1x^136

The gray image is a linear code over GF(2) with n=356, k=13 and d=156.
This code was found by Heurico 1.16 in 8.92 seconds.