The generator matrix

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

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+90x^78+189x^80+28x^81+299x^82+60x^83+374x^84+156x^85+428x^86+284x^87+447x^88+244x^89+410x^90+148x^91+332x^92+84x^93+177x^94+20x^95+138x^96+89x^98+36x^100+32x^102+14x^104+10x^106+2x^108+1x^110+2x^112+1x^136

The gray image is a linear code over GF(2) with n=352, k=12 and d=156.
This code was found by Heurico 1.16 in 2.47 seconds.