The generator matrix

 1  0  0  0  1  1  1 X^2  1  1  1  X  1  X  0  1  1 X^2  X X^2+X  1  1  X X^2 X^2  0  1 X^2+X X^2  1  1  1  1  1  1  1  1  X  X  1  1  X  1  1 X^2  0  1 X^2  1  1  0  X  1  1 X^2  1  0  1 X^2  1  1  1  X X^2+X  1  1  1  1  0  1  1 X^2 X^2+X  1 X^2+X  1  0  1  X  X  1  1 X^2  X  X  1
 0  1  0  0  0  0 X^2  1 X^2+X+1 X+1  1  0 X^2+1  1  1 X^2+X X+1 X^2  1  1 X^2+X X^2+1  1 X^2+X X^2+X  1  X  1  X X+1  X  1 X^2+1 X^2 X+1 X^2  1 X^2  1  X X^2+X X^2  1  1  1  X X^2+X+1  0 X^2  0 X^2+X  1 X^2+X X^2+X  1  1  1 X^2+1  1 X+1 X^2+X+1 X^2  1  0 X^2+X  0  X X^2  1 X^2  X  1  1 X+1  1  1  X X+1  0 X^2+X  X X^2  1 X^2+X  0  0
 0  0  1  0  0  1 X+1  1  1 X^2  0  1 X^2+1 X^2+X+1  0  X X+1  X X^2 X^2+X X^2+X X^2+1 X+1  1  1 X+1 X+1 X+1  1  0 X+1  X X^2+X+1 X^2+1 X^2+X+1  1 X^2+X X^2 X^2+1  0 X^2  0  1 X^2+X X^2  1  X  1 X^2+X X^2+1  1 X^2+X X^2+X+1 X^2+X  X X^2+1  0  X  X  X X+1  X X^2+X+1 X^2+X X+1  0  1  0 X^2+X X^2+1  1 X^2 X^2+1 X^2+X X^2  X X^2 X+1  1  1  X  0  X  1  0  1
 0  0  0  1  1  1  0 X^2+X+1 X^2+1  X  1 X+1  0 X^2  1 X^2+X X^2+X  1 X^2+1  X  1 X+1 X^2+1  1  0 X^2+X X^2+X+1 X+1 X+1 X^2+X+1 X^2+X+1 X^2+X+1 X^2 X^2+X  1 X^2 X^2+X  1 X^2 X+1 X^2+X  1  X  0 X+1  X X+1 X+1 X+1 X^2+X  0 X^2+X+1 X^2+1 X^2+X+1 X^2 X^2+1 X^2+X X^2+1  1  X X+1 X^2+X  X  1  X X^2+X X+1 X^2+1 X^2+X+1 X+1 X+1  0 X^2+X+1  0  0 X^2  1 X^2+1 X+1 X^2+X X^2+X+1  X X+1  X  1  0
 0  0  0  0  X  0  X  0  X  X X^2  X  0 X^2+X  X X^2  0 X^2  X  0  X X^2+X  0 X^2+X  X  0 X^2+X  X X^2 X^2+X X^2 X^2 X^2+X X^2+X  0 X^2  X  0 X^2+X  0 X^2+X X^2+X X^2  0  0 X^2+X X^2  X  X  X  0  0 X^2  0  0 X^2+X X^2+X X^2 X^2+X X^2+X  0 X^2+X X^2+X X^2 X^2+X  0 X^2+X X^2+X  0 X^2 X^2+X  X X^2  0 X^2+X  X  X  X  0 X^2  X  X X^2 X^2+X  0 X^2+X
 0  0  0  0  0 X^2  0  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2  0  0  0  0 X^2 X^2  0  0 X^2  0  0 X^2 X^2  0  0 X^2 X^2 X^2 X^2  0  0 X^2  0  0  0 X^2  0  0  0 X^2 X^2 X^2 X^2  0  0 X^2 X^2  0  0  0 X^2  0 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0  0 X^2 X^2  0  0  0 X^2  0 X^2 X^2 X^2  0 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+243x^76+422x^77+959x^78+956x^79+1612x^80+1694x^81+2235x^82+1950x^83+2816x^84+2248x^85+2855x^86+2178x^87+2890x^88+1964x^89+2195x^90+1544x^91+1351x^92+830x^93+749x^94+376x^95+355x^96+122x^97+118x^98+34x^99+40x^100+12x^101+4x^102+2x^103+2x^104+4x^105+4x^106+2x^108+1x^110

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 61.5 seconds.