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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+42x^69+27x^70+72x^71+64x^72+86x^73+66x^74+208x^75+66x^76+378x^77+90x^78+380x^79+73x^80+184x^81+43x^82+60x^83+34x^84+58x^85+17x^86+44x^87+13x^88+18x^89+10x^90+4x^91+4x^92+2x^93+2x^94+1x^98+1x^136

The gray image is a linear code over GF(2) with n=312, k=11 and d=138.
This code was found by Heurico 1.16 in 2.45 seconds.