The generator matrix

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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+52x^83+324x^84+256x^85+377x^86+244x^87+320x^88+120x^89+157x^90+60x^91+70x^92+28x^93+20x^94+4x^95+5x^96+4x^97+4x^98+1x^110+1x^130

The gray image is a linear code over GF(2) with n=696, k=11 and d=332.
This code was found by Heurico 1.16 in 1.8 seconds.