The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+654x^79+1780x^80+3216x^81+4144x^82+6304x^83+5946x^84+7794x^85+6622x^86+7972x^87+6077x^88+5608x^89+3568x^90+2776x^91+1530x^92+856x^93+279x^94+208x^95+109x^96+40x^97+18x^98+20x^99+4x^100+6x^101+1x^102+2x^103+1x^104

The gray image is a linear code over GF(2) with n=688, k=16 and d=316.
This code was found by Heurico 1.16 in 46.4 seconds.