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  X  0  1  1  1  1  X  1  1  1  0  1  1  1  X  1  1  0  1  1  1  X  1  0  X  1  1  1  X  0  0  1  1  1  1  0 X^2  1  X  X  X  1  0  1  1  1  X  X
 0  X  0 X^2+X  0 X^2+X  0 X^2+X  0 X^2+X X^2 X^2+X  0 X^2+X  X X^2 X^2+X  0 X^2  X  0  X X^2 X^2+X  0 X^2  0 X^2 X^2+X X^2+X  X  X  0 X^2+X  0 X^2+X  0 X^2 X^2+X  X X^2+X X^2+X  X X^2+X X^2+X  X X^2+X  0  X X^2+X X^2  0 X^2+X  0 X^2+X  X X^2 X^2  0 X^2+X X^2  X X^2+X  X X^2  0 X^2+X  X  X  0  X X^2  0  X  0 X^2+X X^2+X  X  X  X  X X^2+X  0  0 X^2+X X^2+X
 0  0 X^2  0  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  0 X^2  0 X^2 X^2  0  0 X^2 X^2  0  0  0  0 X^2  0  0  0  0 X^2 X^2  0 X^2 X^2  0 X^2  0 X^2 X^2  0 X^2  0  0  0 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2  0 X^2 X^2 X^2  0  0  0 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2  0  0  0  0  0  0 X^2
 0  0  0 X^2  0  0  0  0  0  0  0 X^2  0  0 X^2  0 X^2  0  0 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2  0 X^2  0 X^2  0  0 X^2 X^2  0 X^2 X^2  0  0 X^2  0 X^2 X^2 X^2  0 X^2 X^2  0  0  0  0  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0 X^2  0  0 X^2 X^2  0  0 X^2  0 X^2  0 X^2 X^2
 0  0  0  0 X^2  0  0  0  0  0  0  0 X^2  0  0  0  0 X^2  0  0 X^2  0  0  0 X^2 X^2 X^2 X^2  0  0  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 X^2 X^2 X^2  0  0 X^2 X^2  0 X^2 X^2  0 X^2  0  0  0  0 X^2  0 X^2  0  0 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0 X^2 X^2
 0  0  0  0  0 X^2  0  0  0 X^2  0 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2  0  0 X^2 X^2  0  0 X^2  0  0 X^2 X^2  0 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0  0  0  0  0  0 X^2 X^2  0 X^2 X^2  0  0  0  0 X^2 X^2  0 X^2  0  0  0 X^2 X^2  0  0 X^2  0 X^2 X^2 X^2  0  0  0 X^2
 0  0  0  0  0  0 X^2  0  0  0  0  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2  0  0 X^2 X^2 X^2  0  0 X^2  0  0 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2 X^2  0  0 X^2  0 X^2 X^2  0  0  0  0 X^2 X^2  0  0  0  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0  0  0 X^2  0  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2
 0  0  0  0  0  0  0 X^2  0 X^2  0  0  0  0 X^2  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2  0  0 X^2  0  0 X^2 X^2 X^2  0  0  0  0 X^2  0 X^2  0  0  0 X^2 X^2  0  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2 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  0  0 X^2  0 X^2  0 X^2  0  0  0 X^2  0 X^2  0  0
 0  0  0  0  0  0  0  0 X^2  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0  0  0 X^2  0 X^2  0  0 X^2 X^2  0 X^2 X^2  0  0 X^2 X^2  0 X^2 X^2  0 X^2  0 X^2  0  0  0  0  0  0  0  0 X^2 X^2  0  0  0  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2  0 X^2 X^2  0  0  0 X^2 X^2  0 X^2 X^2 X^2  0  0  0  0  0 X^2 X^2 X^2  0  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+144x^76+120x^78+20x^79+348x^80+80x^81+278x^82+160x^83+511x^84+240x^85+396x^86+264x^87+386x^88+176x^89+344x^90+64x^91+260x^92+16x^93+124x^94+4x^95+94x^96+18x^98+22x^100+16x^104+6x^108+3x^112+1x^132

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 3.7 seconds.