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

generates a code of length 89 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+628x^82+1842x^83+3525x^84+4660x^85+5571x^86+6334x^87+6954x^88+7142x^89+7161x^90+6558x^91+5435x^92+3968x^93+2371x^94+1498x^95+1040x^96+414x^97+287x^98+60x^99+36x^100+24x^101+14x^102+8x^103+1x^104+4x^107

The gray image is a linear code over GF(2) with n=712, k=16 and d=328.
This code was found by Heurico 1.16 in 53 seconds.