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

generates a code of length 83 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+114x^80+248x^82+256x^83+352x^84+8x^86+44x^88+1x^160

The gray image is a linear code over GF(2) with n=664, k=10 and d=320.
This code was found by Heurico 1.16 in 1.3 seconds.