The generator matrix

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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 75.

Homogenous weight enumerator: w(x)=1x^0+192x^75+665x^76+1022x^77+1579x^78+2260x^79+2635x^80+3332x^81+3920x^82+4300x^83+4966x^84+5154x^85+5425x^86+5472x^87+4819x^88+4414x^89+4023x^90+3476x^91+2555x^92+1816x^93+1400x^94+920x^95+520x^96+310x^97+190x^98+80x^99+58x^100+16x^101+4x^102+4x^103+5x^104+2x^106+1x^130

The gray image is a linear code over GF(2) with n=344, k=16 and d=150.
This code was found by Heurico 1.13 in 72.1 seconds.