The generator matrix

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

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+144x^68+56x^69+272x^70+280x^71+670x^72+408x^73+746x^74+488x^75+786x^76+568x^77+836x^78+552x^79+727x^80+424x^81+454x^82+216x^83+268x^84+80x^85+90x^86+59x^88+24x^90+24x^92+10x^94+6x^96+2x^100+1x^104

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 5.28 seconds.