The generator matrix

 1  0  0  0  1  1  1  1 X^3+X^2  1 X^2+X  1 X^3  0  1 X^3+X  1  1 X^3  X  1  X  1  0  1  1  1 X^3+X^2  1  1 X^2+X X^3  X  1  1 X^3+X  1  1  1 X^2+X  1  0 X^3+X  1 X^2+X  1 X^3+X^2+X X^3  1  1 X^2+X  1 X^2  1  1  1  1 X^3+X^2+X  1 X^3+X^2 X^3+X^2  1  1  1 X^2 X^2+X  1  1  1 X^3+X^2 X^3  1  1 X^3+X X^3+X^2+X  1 X^3  1  1 X^3  1  1  1 X^2+X  1 X^3+X^2 X^3+X^2  1
 0  1  0  0  X X^2+1 X^3+1 X^2  1 X^3+X+1  1 X^3+X^2+X  1  X X^3+X+1  1 X+1 X^2+X  1  1  1 X^3+X^2+X X^3+1  1 X^3+X X^2+X X^2+X X^3 X^3+X^2+X+1 X^3+X^2 X^3+X^2  1  1 X^3+X+1 X^3  1 X^3+X X+1 X^3+X^2+1 X^3+X^2+X X^3  0  0 X^3+X+1  1 X^2+1  1  1 X^2 X^2+X+1  1  1 X^2+X X^3+X X^3+X+1 X^3 X^2+X+1 X^2  X  X  0 X+1 X^3+1 X^2+X  1  X X^3+X+1  1 X^2+X X^2  1 X^2+X X^3+X^2+X+1  1  1 X^3+X^2 X^2 X^3+1 X^3 X^2 X+1 X^3+X^2 X^3+1 X^3  1  1 X^3+X  0
 0  0  1  0  0 X^2  1 X^2+1  1 X^2+1 X^2+X+1 X^3+1 X^2+X  1 X^2+X X^2+1 X+1 X^2+X+1 X^3+X^2 X^3+1 X^3+X+1  X X^3+X^2+X X^2+X X^2+1 X^3+X^2 X^3+X  1 X^2 X^3+X^2  1 X^3+X^2+X+1 X^2+X+1  1 X^2 X^3+X X^2+X+1 X^3+X^2+X X^2+X+1  1 X^2+X+1  1 X^3+X^2+X X^3+X  X X^3+X^2 X^2+1 X^2+1  X X^3+X^2+1 X^3+X  0  1 X^2 X^2 X^3+X^2 X^3  1 X^3+X^2+X+1  1 X^2  0 X+1  1 X^2+X  1 X^3 X^3+1 X^2+X+1  1 X^3+X^2+X+1 X^3+X^2+1 X^2+X X^3+X^2 X^3+X^2 X^3+X  1 X^3 X^3+1  1 X^3+X^2+X+1 X^3+X^2+X X^2+X  1 X^3+X^2+X+1 X^3+X^2+1  1  0
 0  0  0  1  1 X^2+X+1 X^2 X^3+X^2+X+1 X^2+X+1 X^2+1  0 X^3+X X+1 X^3+X+1 X^3 X^3+X^2+X+1 X^2 X^3+X X+1 X^3+X^2  1  1 X^3+1 X^2+X X^2+1  0  1 X^2+X  0 X^3+X+1 X^3+X^2+X+1  0 X^2+X+1 X^3+X^2+X+1 X^3+X^2+X X^3+X^2+X+1 X^3+X^2+X+1 X^3  X X^3+X X^2+X  1  1 X+1 X^3+X  0  1 X^3 X+1 X^3+1 X^2+X+1  1 X^3  X X^3+X^2+X  1 X^3+1  1 X^2+X+1 X^2  1 X^3+X+1 X^3+X^2+X X^3+X+1 X^2 X^2 X^2 X^3+X+1  0 X^2+X+1 X^3+X^2+X X^3+X^2 X^3+X^2+1 X^3+X^2+X  0 X^3+X^2+1 X^2+X X^3+X X^2+X+1 X^3+X+1 X+1 X^2+X+1  0 X^2+X X^2 X^3 X^3+X^2  0
 0  0  0  0 X^2  0  0  0  0 X^2 X^2 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^2 X^3 X^3 X^3 X^3 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2  0 X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2  0 X^3+X^2 X^3  0 X^3 X^3  0 X^3 X^2 X^3+X^2 X^3 X^2 X^3  0 X^3+X^2 X^3+X^2  0  0  0 X^3+X^2  0  0 X^3 X^3+X^2 X^3 X^2 X^3+X^2 X^3 X^3 X^2  0  0 X^3 X^3  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^2  0 X^3+X^2 X^3+X^2 X^2  0 X^2 X^3 X^3+X^2 X^3+X^2  0

generates a code of length 88 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+62x^78+792x^79+1886x^80+4050x^81+6602x^82+10692x^83+15332x^84+20614x^85+25509x^86+29584x^87+31337x^88+29780x^89+26738x^90+21180x^91+14852x^92+10134x^93+6128x^94+3576x^95+1673x^96+938x^97+344x^98+176x^99+62x^100+50x^101+19x^102+8x^103+7x^104+4x^106+8x^107+2x^108+2x^109+2x^110

The gray image is a linear code over GF(2) with n=704, k=18 and d=312.
This code was found by Heurico 1.16 in 836 seconds.