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

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+34x^81+93x^82+144x^83+313x^84+270x^85+376x^86+364x^87+215x^88+34x^89+106x^90+36x^91+47x^92+14x^93+1x^162

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 0.813 seconds.