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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+128x^79+62x^80+640x^81+62x^82+128x^83+1x^96+1x^98+1x^130

The gray image is a linear code over GF(2) with n=648, k=10 and d=316.
This code was found by Heurico 1.16 in 4.14 seconds.