The generator matrix

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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+436x^84+152x^85+468x^86+104x^87+385x^88+104x^89+280x^90+24x^91+72x^92+4x^94+8x^96+8x^100+1x^120+1x^128

The gray image is a linear code over GF(2) with n=696, k=11 and d=336.
This code was found by Heurico 1.16 in 2.12 seconds.