The generator matrix

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

generates a code of length 89 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+70x^78+260x^79+563x^80+952x^81+915x^82+1742x^83+1512x^84+2270x^85+1923x^86+2736x^87+2278x^88+2798x^89+2140x^90+2740x^91+2012x^92+2124x^93+1384x^94+1428x^95+832x^96+812x^97+489x^98+404x^99+146x^100+118x^101+45x^102+30x^103+14x^104+14x^105+8x^106+2x^107+2x^108+2x^110+2x^111

The gray image is a linear code over GF(2) with n=356, k=15 and d=156.
This code was found by Heurico 1.16 in 62.3 seconds.