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

generates a code of length 97 over Z2[X]/(X^3) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+146x^86+353x^88+24x^89+496x^90+144x^91+599x^92+288x^93+718x^94+352x^95+844x^96+424x^97+918x^98+368x^99+680x^100+304x^101+456x^102+96x^103+358x^104+48x^105+228x^106+165x^108+88x^110+55x^112+20x^114+11x^116+5x^120+2x^122+1x^148

The gray image is a linear code over GF(2) with n=388, k=13 and d=172.
This code was found by Heurico 1.16 in 9.58 seconds.