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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 73.

Homogenous weight enumerator: w(x)=1x^0+46x^73+132x^74+134x^75+282x^76+306x^77+418x^78+500x^79+593x^80+782x^81+908x^82+1010x^83+1134x^84+1318x^85+1305x^86+1368x^87+1209x^88+1056x^89+844x^90+750x^91+598x^92+426x^93+340x^94+248x^95+187x^96+130x^97+124x^98+74x^99+79x^100+30x^101+14x^102+12x^103+9x^104+2x^105+8x^106+2x^108+2x^110+1x^112+1x^116+1x^118

The gray image is a linear code over GF(2) with n=344, k=14 and d=146.
This code was found by Heurico 1.16 in 30.8 seconds.