The generator matrix

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

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+152x^76+494x^77+895x^78+1194x^79+1607x^80+1834x^81+1906x^82+2230x^83+2272x^84+2454x^85+2657x^86+2670x^87+2496x^88+2200x^89+1988x^90+1654x^91+1384x^92+1012x^93+654x^94+382x^95+244x^96+186x^97+81x^98+56x^99+36x^100+12x^101+10x^102+6x^103+1x^106

The gray image is a linear code over GF(2) with n=344, k=15 and d=152.
This code was found by Heurico 1.16 in 55.9 seconds.