The generator matrix

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

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+208x^68+24x^69+566x^70+248x^71+968x^72+320x^73+1444x^74+644x^75+2007x^76+796x^77+2092x^78+820x^79+1988x^80+684x^81+1386x^82+300x^83+837x^84+220x^85+416x^86+36x^87+199x^88+4x^89+90x^90+47x^92+22x^94+11x^96+5x^100+1x^104

The gray image is a linear code over GF(2) with n=312, k=14 and d=136.
This code was found by Heurico 1.16 in 45.7 seconds.