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

generates a code of length 86 over Z2[X]/(X^3) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+35x^76+66x^77+106x^78+138x^79+190x^80+212x^81+262x^82+314x^83+333x^84+334x^85+325x^86+322x^87+263x^88+302x^89+208x^90+174x^91+138x^92+74x^93+87x^94+46x^95+42x^96+28x^97+28x^98+22x^99+16x^100+6x^101+6x^102+6x^103+4x^104+2x^105+1x^106+2x^107+2x^108+1x^130

The gray image is a linear code over GF(2) with n=344, k=12 and d=152.
This code was found by Heurico 1.16 in 2.08 seconds.