The generator matrix

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

generates a code of length 77 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+83x^68+166x^69+212x^70+490x^71+321x^72+680x^73+485x^74+818x^75+452x^76+914x^77+506x^78+824x^79+472x^80+640x^81+239x^82+354x^83+155x^84+130x^85+69x^86+48x^87+37x^88+24x^89+19x^90+20x^91+14x^92+6x^93+4x^94+6x^95+1x^96+1x^98+1x^102

The gray image is a linear code over GF(2) with n=308, k=13 and d=136.
This code was found by Heurico 1.16 in 4.94 seconds.