The generator matrix

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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 71.

Homogenous weight enumerator: w(x)=1x^0+300x^71+416x^72+630x^73+547x^74+786x^75+717x^76+766x^77+550x^78+682x^79+501x^80+536x^81+352x^82+440x^83+274x^84+264x^85+110x^86+136x^87+74x^88+42x^89+37x^90+22x^91+1x^92+2x^93+4x^94+2x^95

The gray image is a linear code over GF(2) with n=312, k=13 and d=142.
This code was found by Heurico 1.16 in 24.6 seconds.