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

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+34x^78+66x^79+111x^80+106x^81+99x^82+114x^83+359x^84+198x^85+138x^86+194x^87+90x^88+144x^89+64x^90+98x^91+48x^92+46x^93+26x^94+36x^95+20x^96+12x^97+20x^98+4x^99+9x^100+4x^101+2x^102+2x^104+2x^105+1x^146

The gray image is a linear code over GF(2) with n=344, k=11 and d=156.
This code was found by Heurico 1.16 in 0.889 seconds.