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

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+74x^69+109x^70+136x^71+164x^72+202x^73+217x^74+316x^75+323x^76+350x^77+464x^78+342x^79+367x^80+286x^81+148x^82+90x^83+106x^84+120x^85+67x^86+54x^87+50x^88+48x^89+19x^90+18x^91+10x^92+8x^93+4x^95+2x^96+1x^124

The gray image is a linear code over GF(2) with n=312, k=12 and d=138.
This code was found by Heurico 1.16 in 26.1 seconds.