The generator matrix

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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+47x^68+70x^69+158x^70+246x^71+345x^72+442x^73+545x^74+622x^75+661x^76+686x^77+688x^78+712x^79+650x^80+622x^81+481x^82+380x^83+286x^84+186x^85+121x^86+78x^87+41x^88+38x^89+31x^90+6x^91+13x^92+2x^93+14x^94+4x^95+3x^96+2x^97+7x^98+1x^100+3x^102

The gray image is a linear code over GF(2) with n=312, k=13 and d=136.
This code was found by Heurico 1.16 in 5.41 seconds.