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  1  X  1  1  X X^2+X  1  1  1  1  1  1  1  1 X^2  1  1 X^2+X  X  0  1  1  0  0  X  1  X  1  1  1  1  1  1  1  1  1 X^2+X  1  1  1  1 X^2+X X^2  0  1  1  X  1  1  X  X  1  1  X  X  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  X  1  1 X^2  1  1 X^2+X+1  0 X^2+X+1 X^2+1 X^2+X+1 X^2 X+1 X^2+X+1  1 X^2  X  1  1  1 X+1 X^2  1  1  1  X  1 X^2+X X+1  0 X^2+X  1  0 X^2+1 X^2+X X^2+X+1  1  1 X^2+X X^2+X  1  1  0  1  X X^2+1  1  0  0  0  X X+1  0 X^2  1  X X^2+X+1
 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 X^2+X X^2+X  0 X^2 X^2+X  0 X^2 X^2  0 X^2 X^2+X  X X^2+X X^2  0 X^2  X X^2+X  0  0  0 X^2  X  X X^2 X^2  0 X^2+X X^2+X X^2+X  0  X  X  0 X^2+X X^2  X  X X^2+X  0  X X^2  X X^2+X X^2+X  0 X^2+X  0  X  0  X X^2+X X^2+X X^2 X^2+X
 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 X^2+X X^2  0 X^2  0 X^2 X^2 X^2 X^2 X^2  X X^2+X  0  X  X  0 X^2  X  0 X^2  0 X^2+X  X  0 X^2+X X^2 X^2+X  0  X X^2+X X^2 X^2  X X^2 X^2+X  X  X X^2 X^2 X^2 X^2 X^2+X X^2+X  X  0  0 X^2  X  X  X  X X^2+X  X X^2  X  X X^2+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  0  0 X^2 X^2  0  0 X^2 X^2 X^2 X^2  0  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2 X^2  0  0 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2  0  0 X^2 X^2  0 X^2 X^2  0 X^2  0  0  0 X^2  0  0  0  0 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 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  0 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2  0  0  0 X^2 X^2 X^2  0 X^2  0  0  0  0  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 X^2  0 X^2 X^2  0 X^2  0 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2  0 X^2  0 X^2 X^2  0  0  0 X^2 X^2 X^2 X^2  0  0 X^2  0 X^2  0 X^2 X^2  0  0  0  0 X^2 X^2  0  0 X^2  0 X^2  0 X^2 X^2  0 X^2  0  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+110x^77+214x^78+314x^79+273x^80+590x^81+486x^82+708x^83+324x^84+868x^85+550x^86+916x^87+354x^88+728x^89+481x^90+470x^91+136x^92+216x^93+127x^94+118x^95+53x^96+32x^97+48x^98+26x^99+4x^100+14x^101+4x^102+2x^103+6x^104+2x^105+9x^106+4x^107+1x^110+2x^111+1x^112

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 82.5 seconds.