The generator matrix

 1  0  0  0  1  1  1  X  1  X  1  1  1  X X^2  1  0 X^2+X  1  1 X^2+X X^2 X^2  1  0  1  1  1  X  1  1  1  1  X  1  0  1  1  0  1  1 X^2  1  X  0  1  1  1  1  1  X  1  X  1  1 X^2+X  1  1  0 X^2  1  1  1  1 X^2  1  0  0 X^2+X X^2 X^2  0  1  1 X^2+X X^2+X X^2+X  X X^2+X  1  1  X X^2 X^2+X  X  1 X^2  1
 0  1  0  0  X  0  X  X X^2+X+1  1 X^2+X+1  1 X^2+X+1  1  1 X^2+X  0  1  1 X^2+1  1 X^2+X  1  X  1 X^2  0 X^2+X  X X^2+X+1 X+1 X^2  1 X^2+X X+1  1 X^2+X+1 X^2 X^2+X  1  0  1  X  1  1 X^2 X^2  0  1 X+1  1 X^2+X X^2+X X^2+1  0  1 X^2+X+1 X^2 X^2 X^2+X  1  X X^2+1 X^2+X+1  1  X  1  1  1  X  1  1 X+1 X^2+X+1  1 X^2 X^2  1 X^2  X X+1 X^2+X  1  1  1  0  X X^2+1
 0  0  1  0  X  1 X+1  1 X^2+X+1 X+1  X X^2+1 X^2  0 X+1  1  1 X^2+X X^2+1 X^2  1 X^2  1 X+1  X  1  X  0  1 X^2+1 X^2+X  X X^2  1  1 X+1 X^2+X+1  1 X^2+X X^2 X^2+1 X^2+X  0  1 X^2+X  0  X  0 X^2+X X+1 X^2+X+1  1  0  0 X+1 X^2+X X^2+1 X^2+X  X  1 X^2+X X^2+1 X+1 X+1 X+1 X^2+1 X^2+X+1 X^2+1  X  X X^2 X^2+1  0 X^2+X X^2+X+1  1  1  1  1  X X^2  1 X^2+X X^2  X  X X^2 X^2
 0  0  0  1 X+1  1  X X+1 X^2+1 X^2+X X^2+1  0  X X+1  1 X^2+X+1  X  1  X  X X^2+1  1 X^2  0 X^2 X+1  0 X+1 X^2+1 X+1 X^2+X  1  1 X^2  1  X X^2 X^2+X+1  1 X^2+X+1 X^2+X X^2+1  X  X  X X^2+1  X  X X^2 X+1 X+1  X  1 X^2+1 X^2+1  0  0 X^2+X+1  1  0  X X^2+X+1 X^2+X X^2+X  1  0 X^2 X^2+X X^2+X+1  1  0 X^2+X  1 X^2+1 X^2+X  0 X^2 X^2+X+1  1  X  0  X X+1 X^2+1 X^2  X  1  X
 0  0  0  0 X^2  0  0  0  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2  0  0  0  0 X^2 X^2 X^2 X^2  0 X^2  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0  0 X^2  0 X^2  0 X^2 X^2 X^2  0  0 X^2  0  0  0  0  0  0  0  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2
 0  0  0  0  0 X^2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  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 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  0 X^2  0 X^2 X^2 X^2 X^2 X^2 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 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0 X^2  0 X^2  0  0 X^2 X^2 X^2  0  0  0  0  0 X^2 X^2  0 X^2 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2  0  0  0  0  0  0  0 X^2  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2 X^2 X^2  0  0  0 X^2  0 X^2  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+291x^78+508x^79+1034x^80+900x^81+1704x^82+1584x^83+2168x^84+1852x^85+2637x^86+2232x^87+2873x^88+2384x^89+2737x^90+2148x^91+2162x^92+1400x^93+1517x^94+808x^95+789x^96+348x^97+324x^98+140x^99+102x^100+28x^101+57x^102+4x^103+15x^104+11x^106+8x^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 62.1 seconds.