The generator matrix

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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+248x^81+132x^82+384x^83+109x^84+268x^85+100x^86+220x^87+60x^88+160x^89+52x^90+88x^91+16x^92+60x^93+28x^94+60x^95+1x^96+32x^97+8x^98+16x^99+3x^100+2x^104

The gray image is a linear code over GF(2) with n=344, k=11 and d=162.
This code was found by Heurico 1.16 in 16.2 seconds.