The generator matrix

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

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+168x^78+232x^79+436x^80+552x^81+648x^82+664x^83+656x^84+716x^85+601x^86+692x^87+521x^88+512x^89+436x^90+348x^91+295x^92+224x^93+148x^94+100x^95+99x^96+40x^97+34x^98+12x^99+32x^100+4x^101+11x^102+7x^104+2x^106+1x^108

The gray image is a linear code over GF(2) with n=344, k=13 and d=156.
This code was found by Heurico 1.16 in 5.44 seconds.