The generator matrix

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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+412x^78+236x^79+394x^80+92x^81+406x^82+148x^83+227x^84+20x^85+72x^86+16x^87+12x^88+2x^90+4x^92+4x^94+1x^108+1x^120

The gray image is a linear code over GF(2) with n=648, k=11 and d=312.
This code was found by Heurico 1.16 in 1.83 seconds.