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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+52x^80+102x^82+32x^83+81x^84+480x^85+582x^86+480x^87+64x^88+32x^89+66x^90+54x^92+18x^94+3x^96+1x^164

The gray image is a linear code over GF(2) with n=688, k=11 and d=320.
This code was found by Heurico 1.16 in 0.859 seconds.