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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+56x^82+112x^83+177x^84+96x^85+215x^86+736x^87+220x^88+96x^89+171x^90+112x^91+46x^92+5x^94+3x^96+1x^98+1x^164

The gray image is a linear code over GF(2) with n=696, k=11 and d=328.
This code was found by Heurico 1.16 in 0.844 seconds.