The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+74x^81+50x^82+220x^83+63x^84+352x^85+553x^86+344x^87+53x^88+212x^89+34x^90+76x^91+9x^92+2x^94+2x^96+2x^97+1x^166

The gray image is a linear code over GF(2) with n=688, k=11 and d=324.
This code was found by Heurico 1.16 in 0.859 seconds.