The generator matrix

 1  0  1  1  1 X^2+X  1  1  0  1  1 X^2+X  0  1  1  1 X^2+X  1  1  1  X  0  1  1  1 X^2+X  1  1  1  0  1  1  1  1  1 X^2  1 X^2+X X^2+X  1  1  1  1  0  1  1  1  1  1  0  1  1  1 X^2+X  1  1  1 X^2+X  1  1 X^2+X  1  X  0  1  0 X^2 X^2  1  0  0  1  1 X^2  1  1  1  1
 0  1 X+1 X^2+X  1  1  0 X+1  1 X^2+X X^2+1  1  1 X^2+X X+1  0  1 X^2+1 X^2 X^2+1  1  1 X^2+X X+1  X  1 X^2+X+1  0 X^2+1  1 X^2+X X^2+1  0 X+1 X^2+X+1  1  0  1  1 X^2+X  X X^2+1 X+1  1  0 X^2+X X+1 X^2+1  0  1  X X^2+1  1  1 X^2+1  0 X^2  1 X^2+X X^2+X+1  1 X^2+X  1  1 X+1  1  1  1 X^2+1  1  1  X X+1  X  1  0 X^2+X  0
 0  0 X^2  0  0  0  0  0  0  0  0  0  0  0 X^2  0 X^2 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2  0  0 X^2  0 X^2 X^2 X^2  0 X^2 X^2  0 X^2  0  0 X^2  0  0 X^2 X^2  0  0 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2  0  0  0 X^2 X^2  0 X^2  0  0  0  0  0 X^2  0 X^2  0 X^2 X^2  0 X^2  0 X^2 X^2
 0  0  0 X^2  0  0  0  0  0  0  0  0  0 X^2  0  0  0  0  0  0  0  0  0 X^2  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  0 X^2  0 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 X^2  0  0  0  0  0  0 X^2  0 X^2  0 X^2  0 X^2  0 X^2  0  0
 0  0  0  0 X^2  0  0  0  0  0  0 X^2 X^2  0  0 X^2 X^2  0 X^2 X^2  0  0 X^2  0 X^2  0  0 X^2 X^2  0 X^2 X^2  0  0  0  0 X^2  0 X^2 X^2  0  0  0 X^2 X^2 X^2 X^2  0  0  0 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0 X^2  0  0  0 X^2 X^2 X^2  0 X^2 X^2  0 X^2  0  0 X^2  0  0  0  0
 0  0  0  0  0 X^2  0  0  0  0 X^2 X^2 X^2  0 X^2  0  0 X^2  0  0  0 X^2 X^2  0 X^2  0 X^2  0  0 X^2  0 X^2  0 X^2  0  0 X^2 X^2  0  0 X^2 X^2  0  0  0 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2 X^2  0  0  0  0 X^2 X^2 X^2  0 X^2 X^2  0 X^2  0  0  0  0  0 X^2  0  0 X^2 X^2
 0  0  0  0  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  0  0 X^2 X^2 X^2  0  0  0 X^2 X^2  0  0 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2  0  0 X^2  0  0 X^2  0 X^2 X^2 X^2  0  0  0 X^2 X^2  0 X^2 X^2 X^2 X^2  0 X^2 X^2  0 X^2  0  0 X^2 X^2  0 X^2 X^2  0  0
 0  0  0  0  0  0  0 X^2  0 X^2  0  0  0 X^2 X^2 X^2 X^2  0 X^2  0 X^2 X^2 X^2  0  0  0  0 X^2 X^2 X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0 X^2 X^2  0  0  0  0  0  0 X^2 X^2 X^2  0 X^2 X^2  0  0  0 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2  0  0 X^2  0  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2
 0  0  0  0  0  0  0  0 X^2  0 X^2 X^2  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 X^2 X^2  0 X^2 X^2 X^2  0 X^2  0 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0 X^2  0  0 X^2  0  0  0 X^2  0  0  0 X^2  0  0  0 X^2 X^2 X^2  0 X^2 X^2  0  0  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+110x^68+24x^69+150x^70+144x^71+489x^72+288x^73+610x^74+464x^75+839x^76+624x^77+800x^78+592x^79+832x^80+480x^81+584x^82+304x^83+429x^84+120x^85+154x^86+32x^87+68x^88+6x^90+21x^92+11x^96+9x^100+7x^104

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 5.36 seconds.