The generator matrix

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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+310x^82+68x^83+160x^84+160x^85+644x^86+264x^87+74x^88+324x^90+20x^91+20x^92+2x^106+1x^128

The gray image is a linear code over GF(2) with n=688, k=11 and d=328.
This code was found by Heurico 1.16 in 0.516 seconds.