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

generates a code of length 80 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+124x^68+494x^70+36x^71+728x^72+132x^73+1076x^74+300x^75+1314x^76+664x^77+1580x^78+976x^79+1735x^80+864x^81+1687x^82+608x^83+1311x^84+360x^85+964x^86+108x^87+576x^88+28x^89+336x^90+20x^91+187x^92+130x^94+32x^96+5x^98+7x^100+1x^108

The gray image is a linear code over GF(2) with n=320, k=14 and d=136.
This code was found by Heurico 1.16 in 27.3 seconds.