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

generates a code of length 80 over Z2[X]/(X^4) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+88x^77+122x^78+236x^79+150x^80+232x^81+94x^82+80x^83+8x^84+6x^86+4x^87+2x^90+1x^128

The gray image is a linear code over GF(2) with n=640, k=10 and d=308.
This code was found by Heurico 1.16 in 0.813 seconds.