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 X 0 1 1 0 1 1 1 1 0 1 X 1 1 0 1 1 1 X X^2 1 1 0 1 X^2 1 1 X 1 1 X X 1 X 1 X 1 X X 1 1 X^2 1 1 X 0 1 X X 1
0 X 0 0 0 0 0 0 X^2 X X^2+X X^2+X X^2+X X^2+X X^2+X X X^2+X X^2 X^2+X X^2 X 0 X^2 X^2+X 0 X X^2+X X X^2 X^2 X^2+X 0 X 0 0 X^2+X X^2 X X X X X X^2 0 0 X X X X X^2+X X X^2+X 0 X^2+X X X X^2 X^2+X X X^2+X X X 0 X^2 X X^2+X X^2 X^2 X X X X^2+X 0 X^2 X^2+X X^2+X 0 X X 0 X^2 X 0 0 0 X
0 0 X 0 0 0 X X^2+X X^2+X X X X^2 X^2+X X 0 X^2 0 X^2 X 0 0 X X^2 X^2+X X^2+X X^2+X 0 0 X^2 X^2+X X X X^2+X X^2 X^2+X X^2+X X^2+X X^2 X^2+X X X^2+X X 0 X 0 0 X^2 X^2+X X^2 0 0 0 X X^2 X^2+X X X^2+X X X X^2+X 0 X^2 X^2+X X^2 X X^2+X X 0 X^2 0 X^2 X^2+X X^2+X X X^2 0 X^2+X X 0 X^2 0 X X^2 X^2+X X^2+X X^2
0 0 0 X 0 X X X 0 X^2 0 X X^2+X X X X^2 X^2 X^2+X 0 X^2 X X^2+X X^2 X^2+X 0 X^2 0 X X 0 X X X^2+X X^2+X X^2+X X X X^2+X 0 0 X X^2+X X X X^2 X^2 0 X X^2+X X^2 X^2 X^2+X 0 0 X^2 X X^2+X X^2 0 X^2+X X^2+X 0 X^2 X^2+X 0 0 X X X^2 X^2+X X^2+X X^2 X X X^2 X^2 X X^2 X X^2 X^2 X^2+X X X^2+X X^2 0
0 0 0 0 X X X^2 X^2+X X X^2 X 0 X X^2 X^2+X X^2+X X^2 X^2+X X 0 X 0 X^2+X X^2 0 0 X^2+X 0 X^2 X^2+X X^2+X X X^2 X^2+X X X X^2 X 0 X^2+X X^2 0 X X^2+X X^2+X 0 X^2+X X 0 X^2 X^2 X^2 X^2 X X X X^2 X 0 0 0 0 X X X^2 0 X^2 X X X X 0 X X X X X 0 X^2+X X^2 X X 0 X^2 0 X^2+X
0 0 0 0 0 X^2 X^2 X^2 0 0 0 X^2 X^2 0 0 0 X^2 0 X^2 X^2 X^2 0 X^2 X^2 X^2 X^2 X^2 0 X^2 X^2 0 0 0 X^2 X^2 0 X^2 0 0 X^2 X^2 X^2 0 0 0 0 0 X^2 X^2 0 X^2 0 X^2 X^2 X^2 0 0 0 0 X^2 0 0 0 X^2 0 X^2 0 0 X^2 X^2 0 X^2 0 0 X^2 X^2 X^2 0 X^2 0 0 0 0 0 0 X^2
generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 77.
Homogenous weight enumerator: w(x)=1x^0+74x^77+152x^78+144x^79+140x^80+208x^81+323x^82+330x^83+301x^84+304x^85+389x^86+370x^87+208x^88+236x^89+232x^90+156x^91+84x^92+98x^93+120x^94+48x^95+24x^96+28x^97+53x^98+34x^99+7x^100+12x^101+11x^102+6x^103+2x^104+1x^128
The gray image is a linear code over GF(2) with n=344, k=12 and d=154.
This code was found by Heurico 1.16 in 58.8 seconds.