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

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+68x^71+136x^72+134x^73+74x^74+176x^75+228x^76+172x^77+224x^78+162x^79+175x^80+144x^81+58x^82+70x^83+64x^84+40x^85+15x^86+26x^87+36x^88+18x^89+11x^90+10x^91+4x^93+1x^94+1x^130

The gray image is a linear code over GF(2) with n=312, k=11 and d=142.
This code was found by Heurico 1.16 in 19.7 seconds.