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

generates a code of length 88 over Z2[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+238x^78+542x^79+879x^80+1240x^81+1326x^82+1768x^83+2105x^84+2324x^85+2329x^86+2366x^87+2717x^88+2528x^89+2352x^90+2280x^91+2082x^92+1624x^93+1261x^94+978x^95+699x^96+464x^97+269x^98+176x^99+80x^100+68x^101+30x^102+18x^103+12x^104+8x^105+1x^106+1x^108+2x^110

The gray image is a linear code over GF(2) with n=352, k=15 and d=156.
This code was found by Heurico 1.16 in 64.4 seconds.