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

generates a code of length 81 over Z2[X]/(X^3) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+47x^68+88x^69+153x^70+190x^71+353x^72+436x^73+501x^74+612x^75+728x^76+920x^77+1077x^78+1176x^79+1249x^80+1328x^81+1287x^82+1304x^83+1078x^84+884x^85+753x^86+554x^87+441x^88+352x^89+237x^90+224x^91+158x^92+76x^93+71x^94+32x^95+27x^96+12x^97+14x^98+4x^99+11x^100+2x^102+2x^108+1x^112+1x^114

The gray image is a linear code over GF(2) with n=324, k=14 and d=136.
This code was found by Heurico 1.16 in 27.9 seconds.