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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+221x^76+8x^77+424x^78+56x^79+583x^80+136x^81+688x^82+316x^83+876x^84+516x^85+794x^86+492x^87+878x^88+324x^89+598x^90+148x^91+394x^92+36x^93+274x^94+12x^95+204x^96+4x^97+92x^98+61x^100+32x^102+13x^104+10x^106+1x^128

The gray image is a linear code over GF(2) with n=344, k=13 and d=152.
This code was found by Heurico 1.16 in 11.3 seconds.