The generator matrix

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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+96x^77+194x^78+320x^79+374x^80+456x^81+540x^82+638x^83+666x^84+592x^85+687x^86+608x^87+644x^88+586x^89+426x^90+384x^91+327x^92+256x^93+131x^94+68x^95+40x^96+44x^97+28x^98+24x^99+14x^100+16x^101+7x^102+2x^103+13x^104+2x^105+2x^106+2x^107+1x^110+2x^111+1x^116

The gray image is a linear code over GF(2) with n=344, k=13 and d=154.
This code was found by Heurico 1.16 in 17.5 seconds.